Enjoy!
Suppose E β [a, b] is infinite. Define F := {x β [a, b] | [a, x] β E is infinte}. F is not empty; in particular, b β F by hypothesis. Clearly F is bounded below by a. Define i := inf F.
Now, let π > 0 be given. Note that for any j β€ i + π, we have that [a, j] β E β [a, i + π] β E. Thus if [a, i + π] β E is finite, then for all j β€ i + π, j β F. In this case, i + π is a lower bound for F, but this contradicts the fact that i is the greatest lower bound of F. Thus [a, i + π] β E is infinite. Next, note that i - π < i, and since i is a lower bound for F, i - π β F. Hence [a, i - π] β E is finite.
Finally, notice that [a, i + π] β E = ([a, i - π] β E) β ((i - π, i + π] β E). Since the left side is infinite and [a, i - π] β E is finite, (i - π, i + π] β E is infinite for all π > 0. Hence i is a limit point of E.
His video is here: https://www.youtube.com/watch?v=g29AFZMUXZI
His proof basically claims that you can construct a Cauchy subsequence of a bounded sequence, and therefore the subsequence converges. However, not all Cauchy sequences converges. Zach's example of a sequence in R happens to be in a complete metric space, which is why the example would converge. However, I also have a hard time wrapping my head around why Cauchy sequences do not converge.
My understanding is that, in a Cauchy sequence, the distance between increasing elements converges to 0. If this is true, why would it not converge. I know this does not contradict the definition of convergence of a sequence, as there is no specific limit point given here, but intuitively I still see Cauchy sequences as converging sequences. Are there any good examples of Cauchy sequences that do not converge?
Hey fellow Redditors :)
I would love some few tips on what to do on new years in Bolzano. Cheers
