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Subsequences

Basically, a subsequence is a sequence with some terms missing. We can make it rigorous.

We can make this rigorous.

Definition

Letbe a sqeuence and letbe any sequence of positive integers such thatThe sequenceis called a subsequence of

The original order of the remaining terms remains. For example if a-n =n^2 thenis a subsequence ofwith every other term deleted.

With subsequences as defined above we can state the following theorems.

Theorem

A sequence converges if and only if each of it's subsequences converges. If every subsequence converges then they and the full sequence all converge to the same limit.

Proof: Assume thatis a sequence that converges toChoose then sinceconverges tothere is a positive integersuch that ifSinceis a subsequence ofandhenceandconverges to A.

does not converge. The subsequenceconverges to 1 and the subsequenceconverges to -1 so both these subsequences converge butifis even and -1 ifis odd. This subsequence does not converge so the original sequence is not convergent.

Theorem

Supposeis a bounded sequence. If all of it's convergent subsequence have the same limit then the sequence converges.

Proof: Assume that all the convergent subsequences have the same limitIfdoes not converge tothen there issuch that for eachthere issuch thatThe sequencebeing bounded will have a convergent subsequence that will converge to something other thancontrary to the condition in the theorem that all convergent subsequences have the same limit. This is a contradiction so the theorem is proved.