Inspirado no

Bolzano-Weierstrass Rap:

https://www.youtube.com/watch?v=eM3S74kchoM
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escrevi

The Bisection Rap
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Come on MS211 students if you wanna be free

from stupid thoughts, then you listen to me

Let f be continuous in [a, b]

and sign(f(a)) be - sign(f(b))

Well, we know that this function has at least one root

and we're thinking to ourselves "This is really good!"

but how to find a root is what we ask

so let's go ahead and deal with this task

Well, are you down with that?

Yo, are you down with that?

Audience: Yes, we are down with that!

(Refrain, can be used several times)

To your right is b and to your left is a

and in between a sequence tries to wind its way

an infinity of x_k's this interval has in it

still you don't know where to look to find the limit

so you stand in the middle halfway in between

let x_0 be (a+b)/2 if you know what I mean

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

to your right another one in the same way lies

and at least one root lies in the other or the one

but kid you'll never believe what this division has done

you see, if a root lies in here

then it is on the right or the left, is this clear?


Well, are you down with that?

Yo, are you down with that?

Audience: Yes, we are down with that!

(Refrain, can be used several times)



So you slide to a side where f changes signs

x_1 is the center of an interval of half the size

this new interval [a_1, b_1] is half as long

contains a zero if my logic ain't wrong

now you do it again, divide the line in two

and if you paid attention, you'll know just what to do

you'll do it k times, things are getting really small

(1/2)^k is your interval

yet with this little space, with this little bound

a root of f can still be found


Well, are you down with that?

Yo, are you down with that?

Audience: Yes, we are down with that!


You can do this forever until Tisha B'Av

cause infinite recursion is what we love

a chain of nested intervals, each inside the last

like little Russian dolls, and they're getting smaller fast



Well, are you down with that?

Yo, are you down with that?

Audience: Yes, we are down with that!



You have to believe and then we're nearly done

that soon b_k - a_k is smaller than epsilon

in this case one often stops

or if the modulus of the midpoint value below epsilon drops



but if we generate intervals [a_k, b_k] on and on

there's exactly one point that lies in every one

see, all those left endpoints they have a supremum 

the same way that the right ones have an infimum

well, this sup and this inf, they lie in each of these sets

though 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 it's a limit so let's call it L!"


Well, are you down with that?

Yo, are you down with that?

Yes, we are down with that!



Recall the kth interval and all the points that it spans

well I only need one, 'cause that's how bad I am

x_k's my name for that point, it lies in interval k

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


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

'cause I got nested intervals and they're working for me

I will come back with an M so big it will do

which is  b - a divided by epsilon  the log base 2


You see the thing with that M is that I picked it so good

that after it the x_k's lie inside of L's hood


L's hood is epsilon sized so all the intervals lie in it

QED, you got a sequence that converges to that limit



Well, are you down with that?

Yo, are you down with that?

Audience: Yes, we are down with that!


























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