Convergence in sequences of sequences
I'm stuck on this problem:
Let be a normed vector space of complex sequences that converge to zero, with the sup-norm:
I need to show that Z is closed in X, but I'm getting a bit confused about sequences of sequences... here's what I got so far:
We will use the fact that Z is closed (in a metric space, as is a NVS) iff it contains all its limit points, i.e.
we want to show that whenever is a sequence in Z, which converges to some then we have that .
So, we take an , which converges to some . By definition, this means that
But how do I use this to show that ?
Possibly by contradiction - suppose is not in Z, then at least one of its members will not satisfy , say, this is the k-th member; so we set and show that this contradicts convergence? But I'm still getting very confused in passing from the infinity-norm to the Euclidean norm...
Is this right, where do I go from here, and is there any way to prove this without contradiction?