Quick questions. A book is asking me to prove the nested interval property assuming only the bolzano-weierstrass theorem.
Is this the proof:
Let [an,bn] be the sequence of intervals
the sequence
a1,a2,a3,... is bounded (easy to prove)
b1,b2,b3,... is bounded (easy to prove)
Thus by the bolzano-weierstrass theorem there exists a subsequence of both an and bn that converges to limit
Pick such a sequence for the a's, say
a1, a54, a666, a878, etc.
And then from the subsequence
b1,b54,b666,b878,... pick a subsequence here that converges (apply bolzano-weierstrass again, just in case)
Then use that sub-subsequence to craft a new. Then say that the limit of the a's is a and the limit of the b's is b. Then the infinite intersection would be [a,b]
But using some set theory we can show that the original intersection of [a1,b1] , [a2,b2, [a3,b3],... is actually equivalent to our new intersection of [a54,b54],[a666,b666], etc.
so this original sequence of closed intervals also converges to [a,b]
Which is always nonempty, even if b=a.
QED?
Is this it? Just imagine that instead of shitty text I bothered to Tex it and that it looks beautiful
