My Name Is ... Bolzano Weierstrass

or

Just Rhymin' With Proof

by essiness

aka Slim Dorky

Andante, with attitude

Come on you math majors if you want to be free

From Corporate America you listen to me.

You've got a sequence that you built from your approximating tweakins

And you really need to find a convergent subsequence,

So you ask my man Bolzano and his homie Weierstrass,

Who've found you a solution with a trick that's really boss.

Well are you down with that?

We're down with that!

Well are you down with that?

We're down with that!

Well you haven't got much hope unless your sequence is bounded,

So let's say some interval has got your numbers surrounded.

They're all greater than a, they're all less than some b,

And it's right there with those points, that you thoughts have to be.

So to your right is b, and to your left is a,

And in between your sequence tries to wind its way,

An infinity of ex-ens, this interval has in it,

Still you don't know where to look, to try to find the limit.

So you stand in the middle, halfway in between,

a plus b over two, if you know what I mean,

To your left a line segment, half the big one's size,

To your right the other half, in the same way lies.

You see every ex-en lives in the other or the one,

But kid you'll never believe what this division has done,

Because if every ex-en lies in one of these, dude,

Then in one or the other an infinitude!

Well are you down with that?

We’re down with that!

Well are you down with that?

We're down with that!

So you slide to the side where this infinity lies,

To the middle of an interval of half the size,

This new interval (I say it's half as long),

Contains an infinite subsequence if my logic ain't wrong.

Now you do it again, divide the line in two,

And if you payed attention, you'll know just what to do.

You count up all the ex-ens, on the left and right,

There's infinity in one, though the space is getting tight.

So you do this k times, now we're really getting small,

One half to the k, is our interval.

Yet in this little space, within this little bound,

A whole subsequence can still be found.

Well are you down with that?

We're down with that!

Well are you down with that?

We're down with that!

Well you can do this forever, until Tishebuv,

Cuz infinite recursion is the thing that we love.

A chain of nested intervals, each inside the last,

Like little Russian dolls, and they're getting smaller fast.

But what you have to believe, because then we're nearly done,

Is there's exactly one point that lives in every one!

See all those left endpoints, they have to have a supremum,

The same way that the right ones have to have an infimum.

Well this sup and this inf, they live in each of these sets,

So the distance that’s between them is as small as it gets.

They are both the same point, so I say what the hell,

I think that its our limit so let's call it L!

Well are you down with that?

We're down with that!

Well are you down with that?

We're down with that!

Well I promised a subsequence and I never tell a lie,

To distinguish it from ex I'll call this sequence why.

Recall the kayth interval, and all the points in its span,

Well I only need one, that's just how bad I am.

Why-kay's my name for this point, it lives in interval kay,

Which makes it quite close to L, you see I planned it that way!

Now you can pick epsilon as small as it wants to be,

Cuz I've got nested quantifiers and their working for me.

I will come back with an M, so big I'm sure it will do,

Which of your epsilon is one minus the log base two.

Well the thing about M, is that I picked it so good,

That after it the why-kays lie inside of L's `hood,

L's `hood is epsilon sized, so all those intervals lie in it,

QED, you've got a sequence that converges to a limit!

Well are you down with that?

We're down with that!

Well are you down with that?

We're down with that!

Well I am outta here now, because my rap is at its end,

But I’ll leave you with this exercise: to prove it in Are-en!