Show that a subsequence of a Cauchy sequence is a Cauchy sequence. The catch is that you can't use the fact that every Cauchy subsequence is convergent.
I really don't know where to begin without assuming what the problem says I can't assume.
Any direction would be appreciated.
Two questions though. Doesn't that statement near complete the proof and also can be the same as in ?
It would seem to me that the subsub scripts would need to be greater then some other number. Maybe I'm being too nitpicky but just to be sure I'm getting what you two are saying I'm going to write it all out.
We start with the known that is convergent and therefore a Cauchy sequence. Then we know from that that there exists some such that .
We then choose some such that the subsequences of and such that
We're then want to prove that
since is convergent it is bounded and therefore so is and .
If we choose a large enough the sequences will be bounded into smaller and smaller intervals because they are bound by the sequences bounds. The process of tightening this window is by choosing larger values of N and therefore forcing larger values of M. This is exactly the Cauchy criterion therefore the subsequence of a Cauchy sequence is a Cauchy sequence.
Does that fly?
The point is: that for Cauchy Sequences it is true that for each there is one that works for then .
I am puzzled by this question.
If the sequence converges then each subsequence converges to the same limit.
Moreover, any convergent sequence is a Cauchy Sequence.
So any subsequence of a Cauchy Sequence must be a Cauchy Sequence.