Real Analysis Definitions

Insert just definitions here, not proofs.

**Convergent Sequence/Limit of a Sequence**: the limit of a sequence is the value that the terms of the sequence ‘tend to’. A sequence is convergent if such a limit exists. Often denoted as

x
n

→
x

{\displaystyle x\_n \to x}

 or

lim

n
→
∞

x
n

=
x

{\displaystyle \lim\_{n \to \infty} x\_n = x}

.

* In

R

{\displaystyle \R}

:

(

x
n

)

{\displaystyle (x\_n)}

 converges to

x

{\displaystyle x}

—or equivalently,

x

{\displaystyle x}

 is the limit of

(

x
n

)

{\displaystyle (x\_n)}

—if

∀
ε
>
0

{\displaystyle \forall \varepsilon>0}

, there exists an

N
∈

N

{\displaystyle N \in \N}

 such that

∀
n
>
N

{\displaystyle \forall n > N}

, we have:

x
−

x

n

<
ε

{\displaystyle |x-x\_{n}|<\varepsilon }

.
* In metric spaces:

(

x
n

)

{\displaystyle (x\_n)}

 converges to a point in the metric space

(
X
,
d
)

{\displaystyle (X,d)}

,

x
∈
X

{\displaystyle x \in X}

—or equivalently,

x
∈
X

{\displaystyle x \in X}

 is the limit of

(

x
n

)

{\displaystyle (x\_n)}

—if

∀
ε
>
0

{\displaystyle \forall \varepsilon>0}

, there exists an

N
∈

N

{\displaystyle N \in \N}

 such that

∀
n
>
N

{\displaystyle \forall n > N}

, we have:

d
(
x
−

x

n

)
<
ε

{\displaystyle d(x-x\_{n})<\varepsilon }

.
* In topological spaces:

(

x
n

)

{\displaystyle (x\_n)}

 converges to a point in the topological space

(
X
,
τ
)

{\displaystyle (X, \tau)}

,

x
∈
X

{\displaystyle x \in X}

—or equivalently,

x
∈
X

{\displaystyle x \in X}

 is the limit of

(

x
n

)

{\displaystyle (x\_n)}

—if for every neighborhood

U

{\displaystyle U}

of

x

{\displaystyle x}

, there exists an

N
∈

N

{\displaystyle N \in \N}

 such that

∀
n
>
N

{\displaystyle \forall n > N}

, we have:

x
n

∈
U

{\displaystyle x\_n \in U}

.
* **Inequality Rule for Real Sequences:** if

(

x

n

)

n

,
(

y

n

)

n

⊆

R

{\displaystyle (x\_{n})\_{n},(y\_{n})\_{n}\subseteq \mathbb {R} }

 and

x

n

≥

y

n

{\displaystyle x\_{n}\geq y\_{n}}

 for all

n
≥
N

{\displaystyle n \geq N}

, then

lim

n
→
∞

x

n

≥

lim

n
→
∞

y

n

{\displaystyle \lim \_{n\to \infty }x\_{n}\geq \lim \_{n\to \infty }y\_{n}}

. In other words, if

x

n

→
x
,

y

n

→
y

{\displaystyle x\_{n}\to x,~y\_{n}\to y}

 and

x

n

≥

y

n

∀
n
≥
N

⟹

x
≥
y
.

{\displaystyle x\_{n}\geq y\_{n}~\forall n\geq N\implies x\geq y.}
* **Infimum**: greatest lower bound. If it exists, it is unique.
* **Supremum**: least upper bound. If it exists, it is unique.

* **Set**: a collection of different/distinct things (elements). Note: sets do not have any concept of order, nor do repeated elements alter the set:

{
1
,
2
# }
{
2
,
1
# }
{
1
,
1
,
2
}

{\displaystyle \{1,2\}=\{2,1\}=\{1,1,2\}}

. The set with no elements in it is the empty set,

# ∅
{
}

{\displaystyle \varnothing = \{\}}

, which is a subset of every set.
* **Open set**:
* **Closed set**:

S

{\displaystyle S}

 is closed if its complement

S

c

{\displaystyle S^{c}}

 is open. Equivalently, if it contains the limit of any convergent sequence in the set. (A bounded set need not contain its boundaries; if it contains none of its boundaries it is open, if it contains all its boundaries, it is closed.)
* **Bounded set**: a subset

S

{\displaystyle S}

 of a metric space is bounded if

∃
r
>
0

{\displaystyle \exists r>0}

 such that

∀
x
,
y
∈
S

{\displaystyle \forall x, y \in S}

, we have

d
(
x
,
y
)
<
r

{\displaystyle d(x,y)<r}

. In the real numbers, a set is bounded if it has upper bound and lower bounds, therefore a set of real numbers is bounded if it is contained in a finite interval.
* **Compact set**: a set

S

{\displaystyle S}

 is compact if and only if it is sequentially compact, or a.k.a. every open cover of

S

{\displaystyle S}

 has a finite subcover. (Note: the Heine-Borel Theorem shows that in Euclidean spaces,

S

{\displaystyle S}

 is closed and bounded

⟺

{\displaystyle \iff}

S

{\displaystyle S}

 is compact).
* **Closure of a set**: the closure of a set

S

{\displaystyle S}

 (denoted

S
¯

{\displaystyle \bar{S}}

) is the intersection of all closed sets containing

S

{\displaystyle S}

. Informally, this is the smallest closed set that contains

S

{\displaystyle S}

. Also,

S
¯

=
S
∪
{

lim

n
→
∞

a

n

:

a

n

∈
A
∀
n
∈

N

}

{\displaystyle {\bar {S}}=S\cup \{\lim \_{n\to \infty }a\_{n}:a\_{n}\in A\forall n\in \mathbb {N} \}}

.
* **Dense set**:

S
⊆
X

{\displaystyle S \subseteq X}

 is dense in

X

{\displaystyle X}

 if

∀
x
∈
X

{\displaystyle \forall x \in X}

,

x
∈
S

{\displaystyle x\in S}

 or

x

{\displaystyle x}

 is ‘arbitrarily close’ to an element in

S

{\displaystyle S}

.
* **Order-dense set**: for any two elements (that have an order defined between them), there is an additional element in between them. A partial or total order,

R

{\displaystyle R}

, on a set

X

{\displaystyle X}

 is dense if

∀
x
,
y
∈
X

{\displaystyle \forall x,y \in X}

 for which

x
R
y

{\displaystyle xRy}

, there

∃
z
∈
X

{\displaystyle \exists z\in X}

 such that

x
R
z
R
y

{\displaystyle xRzRy}

.
* **Power set**: the ‘set’ (*class*/*family)* of all possible subsets of a set, denoted by

2

S

:=
{
X

X
⊆
S
}

{\displaystyle 2^{S}:=\{X|X\subseteq S\}}

.

Functions

* **Continuous function**: Let

(
X
,
d
)

{\displaystyle (X,d)}

 and

(

X
′

,

d
′

)

{\displaystyle (X',d')}

 be metric spaces. A function

f
:
X
→

X
′

{\displaystyle f:X\to X'}

 is continuous at the point

x
0

{\displaystyle x\_0}

 if for any

ε
>
0

{\displaystyle \varepsilon > 0}

, there exists a

δ
>
0

{\displaystyle \delta > 0}

 s.t.

d
(
x
,

x

0

)
<
δ

{\displaystyle d(x,x\_{0})<\delta }

 implies

d
′

(
f
(
x
)
,
f
(

x

0

)
)
<
ε

{\displaystyle d'(f(x),f(x\_{0}))<\varepsilon }

 whenever

x
∈
X

{\displaystyle x \in X}

 . A function is continuous if it is continuous at every point

x
0

∈
X

{\displaystyle x\_0 \in X}

.
* **Graph of a function**: the graph of a function

f
:
X
→
Y

{\displaystyle f: X \to Y}

, is the set

G
r
f
:=
{
(
x
,
f
(
x
)
)

x
∈
X
# }
{
(
x
,
y
)
∈
X
×
Y

# y
f
(
x
)
}

{\displaystyle Grf:=\{(x,f(x))|x\in X\}=\{(x,y)\in X\times Y|y=f(x)\}}

.

Correspondences

* **Graph of a correspondence**: the graph of a correspondence

F
:
X
⇉
Y

{\displaystyle F:X\rightrightarrows Y}

 is the set

G
r
F
:=
{
(
x
,
y
)
∈
X
×
Y

y
∈
F
(
x
)
}

{\displaystyle GrF:=\{(x,y)\in X\times Y|y\in F(x)\}}

. (note that the first definition in the graph of a function no longer works because

f
(
x
)

{\displaystyle f(x)}

 is a set, so that the ordered pair would have a set as the second object).

### Quasiconvex/quasiconcave

A function

f
:
X
→

R

{\displaystyle f: X \to \R}

defined on a convex subset

S

{\displaystyle S}

 of a real vector space is quasiconvex if for all

x
,
y
∈
S

{\displaystyle x, y \in S}

 we have:

* **Convex function**:

f
(
λ
x
(
1
−
λ
)
y
)
≤
λ
f
(
x
)
(
1
−
λ
)
f
(
y
)

{\displaystyle f(\lambda x+(1-\lambda )y)\leq \lambda f(x)+(1-\lambda )f(y)}

, where

λ
∈
[
0
,
1
]

{\displaystyle \lambda \in [0, 1]}
* **Concave function**:

f
(
λ
x
(
1
−
λ
)
y
)
≥
λ
f
(
x
)
(
1
−
λ
)
f
(
y
)

{\displaystyle f(\lambda x+(1-\lambda )y)\geq \lambda f(x)+(1-\lambda )f(y)}

, where

λ
∈
[
0
,
1
]

{\displaystyle \lambda \in [0, 1]}
* **Strictly convex function**:

f
(
λ
x
(
1
−
λ
)
y
)
<
λ
f
(
x
)
(
1
−
λ
)
f
(
y
)

{\displaystyle f(\lambda x+(1-\lambda )y)<\lambda f(x)+(1-\lambda )f(y)}

, where

λ
∈
(
0
,
1
)

{\displaystyle \lambda \in (0, 1)}
* **Strictly concave function**:

f
(
λ
x
(
1
−
λ
)
y
)
>
λ
f
(
x
)
(
1
−
λ
)
f
(
y
)

{\displaystyle f(\lambda x+(1-\lambda )y)>\lambda f(x)+(1-\lambda )f(y)}

, where

λ
∈
(
0
,
1
)

{\displaystyle \lambda \in (0, 1)}
* **Quasiconvex function**:

f
(
λ
x
(
1
−
λ
)
y
)
≤
m
a
x
{
f
(
x
)
,
f
(
y
)
}

{\displaystyle f(\lambda x+(1-\lambda )y)\leq max\{f(x),f(y)\}}

, where

λ
∈
[
0
,
1
]

{\displaystyle \lambda \in [0, 1]}
* **Quasiconcave function**:

f
(
λ
x
(
1
−
λ
)
y
)
≥
m
i
n
{
f
(
x
)
,
f
(
y
)
}

{\displaystyle f(\lambda x+(1-\lambda )y)\geq min\{f(x),f(y)\}}

, where

λ
∈
[
0
,
1
]

{\displaystyle \lambda \in [0, 1]}
* **Strictly quasiconvex function**:

f
(
λ
x
(
1
−
λ
)
y
)
<
m
a
x
{
f
(
x
)
,
f
(
y
)
}

{\displaystyle f(\lambda x+(1-\lambda )y)<max\{f(x),f(y)\}}

, where

λ
∈
(
0
,
1
)

{\displaystyle \lambda \in (0, 1)}
* **Strictly quasiconcave function**:

f
(
λ
x
(
1
−
λ
)
y
)
>
m
i
n
{
f
(
x
)
,
f
(
y
)
}

{\displaystyle f(\lambda x+(1-\lambda )y)>min\{f(x),f(y)\}}

, where

λ
∈
(
0
,
1
)

{\displaystyle \lambda \in (0, 1)}

A function  defined on a convex subset  of a real vector space is quasiconvex if for all  and  we have