Down With Bolzano-Weierstrass

or

Just Rhymin' With Proof

by essiness

        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 x_ns, 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 x_n lives in the other or the one,
But kid you'll never believe what this division has done,
Because if every x_n 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 said 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 x_ns 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 lie 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 x I'll call this sequence y.
Recall the kth interval, and all the points in its span,
Well I only need one, cause that's how bad I am.
y_k's my name for this point, it lies in interval k
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.
You see the thing about M is that I picked it so good,
That after it the y_ks 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 Rⁿ!