Thread: Simple question regarding sequences and subsequences

I'm sure I'm overlooking or misunderstanding some obvious detail here, but nevertheless:

It can be shown that for any sequence that converges to some limit L, any subsequence of also converges to L. Correct? But I don't quite understand this. Suppose we have the (arbitrary) sequence , which converges to 5. But then is a subsequence of , which converges to 3, no? What obvious detail am I missing here?

Also, does every sequence necessarily have a subsequence?

Originally Posted by Ares_D1

I'm sure I'm overlooking or misunderstanding some obvious detail here, but nevertheless:

It can be shown that for any sequence that converges to some limit L, any subsequence of also converges to L. Correct? But I don't quite understand this. Suppose we have the (arbitrary) sequence , which converges to 5. But then is a subsequence of , which converges to 3, no? What obvious detail am I missing here?

Also, does every sequence necessarily have a subsequence?

Dear Ares_D1,

If you consider the definition of the limit of a sequence, it is defined for infinite sequences only.( Limit of a sequence - Wikipedia, the free encyclopedia) Therefore for a finite sequence there is no such thing as a limit.