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, anysubsequenceof 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?