aarjaneiro/planetmath huggingface dataset:
pseudoparadox in measure theory

pseudoparadox in measure theory

The interval

[

0

,

1

)

can be subdivided into an

countably infinite

collection

of disjoint subsets

A

i

,

i

=

0

,

1

,

2

,

…

such that, by translating each set one obtains a collection of

disjoint sets

B

i

such that

⋃

i

=

1

∞

B

i

=

[

0

,

2

)

This

paradox

challenges the naive “” notion that if one a set into a

countable

number of pieces and reassembles them, the result will have the same

measure

as the original set.

The

resolution

to this and

similar

paradoxes lies in the fact that the sets

A

i

were not defined constructively. To show that they exist, one needs to appeal to the non-constructive

axiom of choice

. What the paradox shows is that one can’t have one’s cake and eat it too — either one can cling to the naive “” picture and forego non-constructive techniques as the intuitionist school of mathematics does, or else if, like the majority of mathematicians, one wants to keep the powerful tools provided by non-constructive techniques in

set theory

, one must give up the naive notion that every set is measurable and limit “” to

operations

involving

measurable sets

.

It might be worth mentioning that it is essential that there be an

infinite

number of sets

A

i

. As an elegant

argument

posted by jihema shows, it is not possible to find a finite collection of disjoint subsets of

[

0

,

1

)

such that a union of translations of these subsets equals

[

0

,

2

)

. In higher dimensions, the situation is worse because, as Banach and Tarski showed, it is possible to derive analogous paradoxes involving only a finite number of subsets.

Title

pseudoparadox in measure theory

Canonical name

PseudoparadoxInMeasureTheory

Date of creation

2013-03-22 14:38:40

Last modified on

2013-03-22 14:38:40

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

11

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 28E99

Synonym

one-dimensional Banach-Tarski paradox

Related topic

VitalisTheorem

Related topic

BanachTarskiParadox

Generated on Fri Feb 9 20:11:50 2018 by

LaTeXML
analytic curve

analytic curve

There are several somewhat different definitions of the word

analytic curve

depending on context.
In the context of a

real analytic

manifold

(for example

ℝ

n

), the most

generic

definition is perhaps the following.

Definition

.

Suppose

X

is a real analytic manifold.
A curve

γ

⊂

X

is an

analytic curve

if it is a

real analytic submanifold

of

dimension

1. Equivalently if near each point

p

∈

γ

,

there exists a real analytic mapping

f

:

(

-

1

,

1

)

→

X

,

such that

f

has nonvanishing

differential

and maps onto a

neighbourhood

of

p

in

γ

.

It is sometimes common to equate the mapping

f

and the curve

γ

.
If the curve is as above but instead in the

complex plane

, we can instead make the following

equivalent

definition.

Definition

.

A curve

γ

⊂

ℂ

is said to be an

analytic curve

(or

analytic arc

) if every point of

γ

has an open neighbourhood

Δ

for which there is an
onto

conformal map

f

:

𝔻

→

Δ

(where

𝔻

⊂

ℂ

is
the

unit disc

) such that

𝔻

∩

ℝ

is mapped
onto

Δ

∩

γ

by

f

.

Other words for this concept are

smooth analytic curve

, in which case the word

analytic curve

would be reserved for curves with

singularities

. That is, for

real analytic subvarieties

of

X

. Some authors will emphasize the fact that this is a real curve and say

real analytic curve

.

In the context of

subvarieties

the following definition may be used.

Definition

.

An

analytic curve

is a

complex analytic subvariety

of dimension 1 of a

complex manifold

.

Note that locally all complex analytic subvarieties of dimension 1 in

ℂ

2

can be parametrized by a the

Puiseux parametrization

theorem. Perhaps that is why there is the confusion in using the term.

References

1

Theodore B. Gamelin.
.
Springer-Verlag, New York, New York, 2001.

2

Hassler Whitney.
.
Addison-Wesley, Philippines, 1972.

Title

analytic curve

Canonical name

AnalyticCurve

Date of creation

2013-03-22 14:18:03

Last modified on

2013-03-22 14:18:03

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

7

Author

jirka (4157)

Entry type

Definition

Classification

msc 30-00

Classification

msc 54-00

Synonym

analytic arc

Synonym

smooth analytic curve

Synonym

real analytic curve

Related topic

FreeAnalyticBoundaryArc

Generated on Fri Feb 9 20:12:09 2018 by

LaTeXML
annulus

annulus

An

annulus

is the region

bounded

between two (usually concentric) circles.

An

open annulus

is a domain in the

complex plane

of the form

A

=

A

w

⁢

(

r

,

R

)

=

{

z

∈

ℂ

:

r

<

|

z

-

w

|

<

R

}

,

where

w

is an arbitrary

complex number

, and

r

and

R

are

real numbers

with

0

<

r

<

R

. Such a set is often called an

annular region

.

It should be noted that the annulus usually refers to an open annulus.

More generally, one can allow

r

=

0

or

R

=

∞

. (This makes sense for the purposes of the bound on

|

z

-

w

|

above.) This would make an annulus include the cases of a punctured disc, and some unbounded domains.

Analogously, a

closed annulus

is a set of the form

A

¯

=

A

¯

w

⁢

(

r

,

R

)

=

{

z

∈

ℂ

:

r

≤

|

z

-

w

|

≤

R

}

,

where

w

∈

ℂ

, and

r

and

R

are real numbers with

0

<

r

<

R

.

One can show that two annuli

D

w

⁢

(

r

,

R

)

and

D

w

′

⁢

(

r

′

,

R

′

)

are

conformally equivalent

if and only if

R

/

r

=

R

′

/

r

′

. More generally, the complement of any closed disk in an open disk is conformally equivalent to precisely one annulus of the form

D

0

⁢

(

r

,

1

)

.

Title

annulus

Canonical name

Annulus1

Date of creation

2013-03-22 13:34:52

Last modified on

2013-03-22 13:34:52

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

7

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 30-00

Synonym

open annulus

Synonym

annular region

Related topic

Annulus

Defines

closed annulus

Generated on Fri Feb 9 20:12:11 2018 by

LaTeXML
biquadratic equation

biquadratic equation

A

biquadratic equation

(in a narrower sense) is the special case of the quartic equation (

http://

planetmath

.org/QuarticFormula

) containing no odd degree terms:

a

⁢

x

4

+

b

⁢

x

2

+

c

=

0

(1)

Here,

a

,

b

,

c

are known real or

complex numbers

and

a

≠

0

.

For solving a biquadratic equation (1) one does not need the quartic formula (

http://planetmath.org/QuarticFormula

) since the equation may be thought a

quadratic equation

with respect to

x

2

, i.e.

a

⁢

(

x

2

)

2

+

b

⁢

x

2

+

c

=

0

,

whence

x

2

=

-

b

±

b

2

-

4

⁢

a

⁢

c

2

⁢

a

(see

quadratic formula

or quadratic equation in

ℂ

(

http://planetmath.org/QuadraticEquationInMathbbC

)).  Taking

square roots

of the values of

x

2

(see taking square root algebraically), one obtains the four roots (

http://planetmath.org/Equation

) of (1).

Example.

Solve the biquadratic equation

x

4

+

x

2

-

20

=

0

.

(2)

We have

x

2

=

-

1

±

1

2

-

4

⋅

1

⋅

(

-

20

)

2

⋅

1

=

-

1

±

9

2

,

(3)

i.e.

x

2

=

4

or

x

2

=

-

5

.  The solution is

x

=

±

2

∨

x

=

±

i

⁢

5

.

(4)

Remark.

In one wants to form of

rational numbers

a polynomial equation with rational

coefficients

and most possibly low degree by using two square root operations, then one gets always a biquadratic equation.  A couple of examples:

1)

x

=

1

+

2

+

3

(

x

-

1

)

2

=

2

+

2

⁢

6

+

3

y

2

-

5

=

2

⁢

6

y

4

-

10

⁢

y

2

+

1

=

0

(one has substituted (

http://planetmath.org/TchirnhausTransformations

)

x

-

1

:=

y

)

2)

x

=

2

-

1

x

2

=

2

-

1

(

x

2

+

1

)

2

=

2

x

4

+

2

⁢

x

2

-

1

=

0

Title

biquadratic equation

Canonical name

BiquadraticEquation

Date of creation

2013-03-22 17:52:45

Last modified on

2013-03-22 17:52:45

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

11

Author

pahio (2872)

Entry type

Topic

Classification

msc 30-00

Classification

msc 12D99

Related topic

BiquadraticExtension

Related topic

BiquadraticField

Related topic

EulersDerivationOfTheQuarticFormula

Related topic

IrreduciblePolynomialsObtainedFromBiquadraticFields

Related topic

LogicalOr

Defines

biquadratic equation

Generated on Fri Feb 9 20:12:17 2018 by

LaTeXML
domain

domain

A

http://

planetmath

.org/node/4811

connected

non-empty

open set

in

ℂ

n

is called a

domain

.

The

topology

considered is the

Euclidean

one (viewing

ℂ

as

ℝ

2

). So we have that for a domain

D

being connected is

equivalent

to being

path-connected

.

Since we have that every component of a region

D

will be a domain, we have that every region has at most countably many components.

This

definition

has no particular relationship to the notion of an integral domain (

http://planetmath.org/IntegralDomain

), used in algebra. In number theory, one sometimes talks about fundamental domains in the upper half-plane, these have a different definition and are not normally open. In

set theory

, one often talks about the domain (

http://planetmath.org/

Function

) of a function. This is a separate

concept

. However, when one is interested in complex analysis, it is often reasonable to consider only functions defined on connected open sets in

ℂ

n

, which we have called domains in this entry. In this context, the two notions coincide.

A

domain

in a

metric space

(or more generally in a

topological space

) is a connected open set.

Cf.

http://

mathworld

.wolfram.com/Domain.html

Mathworld,

http://en.

wikipedia

.org/wiki/Domain

Wikipedia.

Title

domain

Canonical

name

Domain

Date of creation

2013-03-22 11:56:17

Last modified on

2013-03-22 11:56:17

Owner

drini (3)

Last modified by

drini (3)

Numerical id

13

Author

drini (3)

Entry type

Definition

Classification

msc 30-00

Classification

msc 54A05

Classification

msc 54E35

Related topic

Region

Related topic

Topology

Related topic

ComplexNumber

Related topic

IntegralDomain

Generated on Fri Feb 9 20:12:18 2018 by

LaTeXML
equality of complex numbers

equality of complex numbers

The equality relation “=” among the is determined as consequence of the definition of the

complex numbers

as elements of the

quotient ring

ℝ

/

(

X

2

+

1

)

, which enables the of the complex numbers as the ordered pairs

(

a

,

b

)

of

real numbers

and also as the sums

a

+

i

⁢

b

where

i

2

=

-

1

.

a

1

+

i

⁢

b

1

=

a

2

+

i

⁢

b

2

⟺

a

1

=

a

2

∧

b

1

=

b

2

(1)

This condition may as well be derived by using the field properties of

ℂ

and the properties of the real numbers:

a

1

+

i

⁢

b

1

=

a

2

+

i

⁢

b

2

⟹

a

2

-

a

1

=

-

i

⁢

(

b

2

-

b

1

)

⟹

(

a

2

-

a

1

)

2

=

-

(

b

2

-

b

1

)

2

⟹

(

a

2

-

a

1

)

2

+

(

b

2

-

b

1

)

2

=

0

⟹

a

2

-

a

1

=

0

,

b

2

-

b

1

=

0

⟹

a

1

=

a

2

,

b

1

=

b

2

The implication in the reverse direction is apparent.

If

a

+

i

⁢

b

≠

0

,  then at least one of the real numbers

a

and

b

differs from 0.  We can set

a

=

r

⁢

cos

⁡

φ

,

b

=

r

⁢

sin

⁡

φ

,

(2)

where

r

is a uniquely determined positive number and

φ

is an angle which is uniquely determined up to an integer

multiple

of

2

⁢

π

.  In fact, the

equations

(2) yield

a

2

+

b

2

=

r

2

⁢

(

cos

2

⁡

φ

+

sin

2

⁡

φ

)

=

r

2

,

whence

r

=

a

2

+

b

2

.

(3)

Thus (2) implies

cos

⁡

φ

=

a

a

2

+

b

2

,

sin

⁡

φ

=

b

a

2

+

b

2

.

(4)

The equations (4) are , since the sum of the squares of their is 1.  So these equations determine the angle

φ

up to a multiple of

2

⁢

π

.  We can write the

Theorem.

Every complex number may be represented in the

polar form

r

⁢

(

cos

⁡

φ

+

i

⁢

sin

⁡

φ

)

,

where

r

is the

modulus

and

φ

the argument of the number.  Two complex numbers are equal if and only if they have equal moduli and, if the numbers do not vanish, their arguments differ by a multiple of

2

⁢

π

.

Title

equality of complex numbers

Canonical name

EqualityOfComplexNumbers

Date of creation

2015-05-09 17:11:34

Last modified on

2015-05-09 17:11:34

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Topic

Classification

msc 30-00

Related topic

ModulusOfAComplexNumber

Related topic

ArgumentOfProductAndQuotient

Related topic

ComplexLogarithm

Generated on Fri Feb 9 20:12:21 2018 by

LaTeXML
extended boundary

extended boundary

Definition

.

Suppose that

ℂ

is the

complex plane

and

G

⊂

ℂ

. The

extended boundary

of

G

denoted

∂

∞

⁡

G

is defined as
the

boundary

of

G

plus the

point at infinity

if in fact

G

is unbounded.

The extended boundary of

G

is really the boundary of

G

in the

extended complex plane

(the

one-point
compactification

of

ℂ

).

References

1

John B. Conway.
.
Springer-Verlag, New York, New York, 1978.

Title

extended boundary

Canonical name

ExtendedBoundary

Date of creation

2013-03-22 14:12:11

Last modified on

2013-03-22 14:12:11

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

7

Author

jirka (4157)

Entry type

Definition

Classification

msc 30-00

Classification

msc 54-00

Related topic

BoundaryInTopology

Generated on Fri Feb 9 20:12:23 2018 by

LaTeXML
free analytic boundary arc

free analytic boundary arc

Definition

.

Let

G

⊂

ℂ

be a region and let

γ

be a

connected

subset of

∂

⁡

G

(

boundary

of

G

), then

γ

is a

free analytic boundary arc

of

G

if for every point

ζ

∈

γ

there is a

neighbourhood

U

of

ζ

and
a

conformal equivalence

h

:

𝔻

→

U

(where

𝔻

is the

unit disc

) such that

h

⁢

(

0

)

=

ζ

,

h

⁢

(

-

1

,

1

)

=

γ

∩

U

and

h

⁢

(

𝔻

+

)

=

G

∩

U

(where

𝔻

+

is all the points
in the unit disc with non-negative

imaginary part

).

References

1

John B. Conway.
.
Springer-Verlag, New York, New York, 1995.

Title

free analytic boundary arc

Canonical name

FreeAnalyticBoundaryArc

Date of creation

2013-03-22 14:18:00

Last modified on

2013-03-22 14:18:00

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

7

Author

jirka (4157)

Entry type

Definition

Classification

msc 30-00

Classification

msc 54-00

Related topic

AnalyticCurve

Generated on Fri Feb 9 20:12:25 2018 by

LaTeXML
fundamental theorems in complex analysis

fundamental theorems in complex analysis

The following is a list of fundamental theorems in the subject of

complex analysis

(single complex variable). If a theorem does not yet appear in the encyclopedia, please consider adding it — Planet Math is a work in progress and some basic results have not yet been entered. Likewise, if some basic theorem has been overlooked in this list, please add it.

•

Cauchy-Riemann equations

•

Cauchy’s

integral

theorem

•

second form of Cauchy integral theorem

•

Morera’s theorem

•

Cauchy’s integral formula

•

Cauchy’s

residue theorem

•

Cauchy’s

argument principle

•

Rouché’s theorem

•

identity theorem of power series

•

rigidity theorem for analytic functions

•

Riemann’s

removable singularity

theorem

•

Casorati-Weierstrass theorem

•

implicit function theorem

for

complex analytic functions

(I gave proofs of this and the next theorem in a posting to a forum and must convert them to an encyclopaedia entry.)

•

inverse function theorem

for complex analytic functions

•

maximal modulus principle

•

Schwarz lemma

•

Liouville’s theorem

•

characterization of

rational functions

•

Weierstrass’ factorization theorem

•

Weierstrass’ criterion of

uniform convergence

•

Mittag-Leffler’s theorem

•

MÃÂ¶bius circle transformation theorem

•

Riemann mapping theorem

•

Gauss’ mean value theorem

•

Schwarz’

reflection principle

•

Harnack’s principle

•

Bloch theorem

•

Picard’s theorem (

http://planetmath.org/PicardsTheorem

)

•

Little Picard theorem

•

Monodromy theorem

•

Runge’s theorem

•

Mergelyan’s theorem

•

Montel’s theorem

•

Marty’s theorem

•

Hurwitz’s theorem

•

Bieberbach’s conjecture

•

Koebe one-fourth theorem

•

Factorization theorem for

H

∞

functions

(

http://planetmath.org/FactorizationTheoremForHinftyFunctions

)

•

Plemelj formulas

•

Harnack theorem

•

Schwarz and Poisson formulas

Title

fundamental theorems in complex analysis

Canonical name

FundamentalTheoremsInComplexAnalysis

Date of creation

2013-03-22 14:57:33

Last modified on

2013-03-22 14:57:33

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

26

Author

rspuzio (6075)

Entry type

Topic

Classification

msc 30-00

Related topic

TopicEntryOnComplexAnalysis

Generated on Fri Feb 9 20:12:26 2018 by

LaTeXML
Heaviside step function

Heaviside step function

The

Heaviside step function

is the

function

H

:

ℝ

→

ℝ

defined as

H

⁢

(

x

)

=

{

0

when

⁢

x

<

0

,

1

/

2

when

⁢

x

=

0

,

1

when

⁢

x

>

0

.

Here, there are many conventions for the value at

x

=

0

. The
motivation for setting

H

⁢

(

0

)

=

1

/

2

is that we can then write

H

as a function of the

signum function

(see
this page (

http://planetmath.org/SignumFunction

)). In applications, such as
the

Laplace transform

, where the

Heaviside function

is used extensively,
the value of

H

⁢

(

0

)

is irrelevant.
The

Fourier transform

of heaviside function is

ℱ

0

⁢

H

⁢

(

t

)

=

1

2

⁢

(

δ

⁢

(

t

)

-

i

π

⁢

t

)

where

δ

denotes the Dirac delta centered at

0

.
The function is named after Oliver Heaviside (1850-1925)

[

1

]

. However, the function was already used by
Cauchy

[

2

]

, who defined the function as

u

⁢

(

t

)

=

1

2

⁢

(

1

+

t

/

t

2

)

and called it a

coefficient

limitateur

[

3

]

.

References

1

The MacTutor History of Mathematics archive,

http://www-gap.dcs.st-and.ac.uk/ history/Mathematicians/Heav
iside.html

Oliver Heaviside.

2

The MacTutor History of Mathematics archive,

http://www-gap.dcs.st-and.ac.uk/ history/Mathematicians/Cauc
hy.html

Augustin Louis Cauchy.

3

R.F. Hoskins,

Generalised functions

,
Ellis Horwood Series: Mathematics and its applications,
John Wiley & Sons, 1979.

Title

Heaviside step function

Canonical

name

HeavisideStepFunction

Date of creation

2013-03-22 13:46:14

Last modified on

2013-03-22 13:46:14

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

8

Author

Koro (127)

Entry type

Definition

Classification

msc 30-00

Classification

msc 26A06

Synonym

Heaviside function

Related topic

SignumFunction

Related topic

DelayTheorem

Related topic

TelegraphEquation

Generated on Fri Feb 9 20:12:28 2018 by

LaTeXML
isolated singularity

isolated singularity

Let

ℂ

∪

{

∞

}

denote the

Riemann sphere

, and let

U

⊂

ℂ

be open.  Let

f

:

U

→

ℂ

∪

{

∞

}

be a

function

.  We say that

z

is an

isolated singularity

of

f

if there exists an

open set

V

⊂

U

containing

z

and such that

f

is

analytic

on

V

∖

{

z

}

.

In other , if we take the set

S

of points in

U

where

f

is

not

analytic, the isolated singularities are exactly the

isolated points

of

S

in the usual topological sense.

There are three kinds of isolated singularities:

•

removable singularities

(

e.g.

⁢

z

=

0

⁢

for the function

⁢

sin

⁡

z

z

)

•

poles

(

e.g.

⁢

z

=

0

⁢

for the function

⁢

1

z

2

)

•

essential singularities

(

e.g.

⁢

z

=

0

⁢

for the function

⁢

exp

⁡

1

z

)

Title

isolated singularity

Canonical name

IsolatedSingularity

Date of creation

2013-03-22 14:01:04

Last modified on

2013-03-22 14:01:04

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

10

Author

bwebste (988)

Entry type

Definition

Classification

msc 30-00

Generated on Fri Feb 9 20:12:31 2018 by

LaTeXML
proof of limit of nth root of n

proof of limit of nth root of n

In this entry, we present a self-contained, elementary
proof of the fact that

lim

n

→

∞

⁡

n

1

/

n

=

1

.
We begin by with inductive proofs of two integer

inequalities

—

real numbers

will not enter until
the very end.

Lemma 1

.

For all integers

n

greater

than or equal to

5

,

2

n

>

n

2

Proof.

We begin with a few easy observations. First,
a bit of

arithmetic

:

2

5

=

32

>

25

=

5

2

Second, some algebraic manipulation of the inequality

n

>

4

:

n

-

1

>

3

(

n

-

1

)

2

>

9

(

n

-

1

)

2

>

2

n

2

-

2

⁢

n

+

1

>

2

2

⁢

n

2

>

n

2

+

2

⁢

n

+

1

2

⁢

n

2

>

(

n

+

1

)

2

These observations provide us with the makings of an
inductive proof. Suppose that

2

n

>

n

2

for some
integer

n

≥

5

. Using the inequality we just showed,

2

n

+

1

=

2

⋅

2

n

>

2

⁢

n

2

>

(

n

+

1

)

2

.

Snce

2

5

>

5

2

and

2

n

>

n

2

implies that

2

n

+

1

>

(

n

+

1

)

2

when

n

≥

5

we conclude that

2

n

>

n

2

dor
all

n

≥

5

.
∎

Lemma 2

.

For all integers

n

greater than or equal to

3

,

n

n

+

1

>

(

n

+

1

)

n

Proof.

We begin by noting that

3

4

=

81

>

64

=

4

3

.

Next, we make assume that

(

n

-

1

)

n

>

n

(

n

-

1

)

.

for some

n

.
Multiplying both sides by

n

:

n

⁢

(

n

-

1

)

n

>

n

n

.

Multiplying both sides by

(

n

+

1

)

n

and making
use of the identity

(

n

+

1

)

⁢

(

n

-

1

)

=

n

2

-

1

,

n

⁢

(

n

2

-

1

)

n

>

n

n

⁢

(

n

+

1

)

n

.

Since

n

2

>

n

2

-

1

, the left-hand side is
less than

n

2

⁢

n

+

1

, hence

n

2

⁢

n

+

1

>

n

n

⁢

(

n

+

1

)

n

.

Canceling

n

n

from both sides,

n

(

n

+

1

)

>

(

n

+

1

)

n

.

Hence, by induction,

n

(

n

+

1

)

>

(

n

+

1

)

n

for all

n

≥

3

.
∎

Theorem 1

.

lim

n

→

∞

⁡

n

1

/

n

=

1

Proof.

Consider the

subsequence

where

n

is a power of

2

.
We then have

(

2

m

)

(

1

/

2

m

)

=

2

m

/

2

m

.

By lemma 1,

m

/

2

m

<

1

/

m

when

m

≥

5

. Hence,

(

2

m

)

1

/

2

m

<

2

1

/

m

. Since

lim

m

→

0

⁡

2

1

/

m

=

1

, and

(

2

m

)

1

/

2

m

)

>

1

, we conclude
by the

squeeze rule

that

lim

m

→

0

⁡

(

2

m

)

1

/

2

m

=

1

.

By lemma 2, the sequence

{

n

1

/

n

}

is

decreasing

. It
is clearly bounded from below by

1

. Above, we exhibited
a subsequence which tends towards

1

. Thus it follows that

lim

n

→

∞

⁡

n

1

/

n

=

1

.

∎

Title

proof of limit of nth root of n

Canonical

name

ProofOfLimitOfNthRootOfN

Date of creation

2014-02-28 7:21:31

Last modified on

2014-02-28 7:21:31

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

19

Author

rspuzio (6075)

Entry type

Proof

Classification

msc 30-00

Classification

msc 12D99

Generated on Fri Feb 9 20:12:37 2018 by

LaTeXML