The proof I know for the Bolzano-Weierstrass Theorem uses Nested Intervals Theorem, for which we can find a infinitely small interval where infinitely many terms can be in that interval.
Now, the book asks me to prove that every sequence $x_n$ of real numbers must have a monotone subsequence. The book says after I prove this, I can discover an easy alternative proof for the Bolzano-Weierstrass Theorem but I can't understand how this is so.
Is it because for bounded sequence, we can find a monotone subsequence that converges to the least upper bound or greatest lower bound? And this is a convergent subsequence?