There is a sequence, <Xn>, such that the subsequence of even subscripts and odd subscripts both converge to x.
How would I show, using this, that the whole sequence <Xn> converges to x?
Let be a sequence. we denote the terms with even subscripts as and those with odd subscripts as . if we let , then the even terms are and the odd terms are
Now, since we have that for all there exists an such that implies . Similarly, for some , we have implies
Now take such an and choose . Then implies that:
That is, we have
Now, how would you continue. (we want to show that the sequence x_n converges. if that happens, then all its subsequences converge to the same limit as the sequence, which in this case is x. Hint: try to show x_n is a Cauchy sequence)