Ah, he's just getting started on his
journey into the set of real numbers!

Eventually he will discover, "God
made the integers. All else is
man made.".

In particular, man made the real
numbers to be complete which
means that every sequence that
appears to converge, that is,
meets, the Cauchy criterion,
actually does converge. Really
his discoveries are about the
completeness property of the
real numbers. So, in particular,
if we have an infinite series
that meets the Cauchy criterion,
then it converges, in particular,
there is real number for it to
converge to. The same statement
is not true in the rational numbers
or the algebraic numbers!

Calculus: Sure, the elementary properties
of the completeness property of the
real numbers.

How do we know that anything like the
real numbers can exist? Because we
can start with the simplest things,
say, just the empty set, do a
lot of set theory pushing around,
construct something that looks like
the natural numbers -- 1, 2, 3, ....
Then we can use the naturals to
construct (something that looks like)
the integers -- ..., -3, -2, -1, 0,
1, 2, 3, ....

Continuing in this way, we can
construct the rationals and the
reals. For the reals, a popular
approach, nicely intuitive, is
Dedekind cuts.

So, we base it all on just
set theory starting with just
the empty set.

So, the reals exist but only because
man said that they exist!

Too soon he will face a danger,
compactness! That's where
every infinite subset has a limit
point, that is, in the infinite
subset is a sequence that converges
to something. This is true if and
only if every open cover has a
finite subcover. And every
closed and bounded subset of
finite dimensional, real Euclidean space is
compact. A real valued continuous
function with domain a compact
set is uniformly continuous
and bounded and achieves both
its upper and lower bounds.
Seeing these results, the poor
guy might lose it! The usual
way we show that the Riemann
integral of calculus exists is
via uniform continuity.

After he recovers from seeing the
completeness property of the
reals, under no circumstances
let him learn about Hilbert space --
a complete inner product space!
Okay, but are there any examples?
Actually, yes: The set of all
real valued random variables
X so that E[X^2] is finite.
Yup, the set of all of these
is complete and, thus, forms
a Hilbert space. Totally mind
blowing that any such thing
could be true! Here complete
means Cauchy convergent means convergent
where we consider distance
in Hilbert space which is the
metric we get from the
inner product. So, with just
that concept of distance,
a Cauchy convergent sequence of E[X^2] finite
random variables actually
converges, that is, there is
a random variable for the sequence
to converge to.
You'd think that random variables
could wiggle too much, but, no,
they can't! Beyond belief.

If he survives these severe
trials of the mind, keep him
away from a classic text on
point set topology, e.g.,
Kelley. There learn that
can have a set A and a point
x so that point x is right
next to set A but there is
no sequence in set A converging
to point x. That is, sequences
are not enough to characterize
the more general case of convergence.
For this more general case, there
is Moore-Smith convergence,
nets, filters, etc.

Somewhere in there he will discover
the continuum hypothesis, model
theory, etc.

But under no circumstances let him
get near

John C. Oxtoby,
'Measure and Category:
A Survey of the Analogies between
Topological and Measure Spaces',
ISBN 3-540-05349-2,
Springer-Verlag,
Berlin,
1971.

If he sees this book, he may never
recover!