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Let F be the subspace of sequences whose first term is zero. Show that F is closed.

Let $((V_{nk}):k=1,2,.....)$ be a convergent sequence in F. I need to show it converges to a sequence whose first term is 0. Well, for all positive integers n, we have

$V_{nk}->x_{n}$ as k tends to infinity. In particular, $V_{n1}->x_{1}$, so I need to show x_{1}=0. Problem is, I don't see why this should be true. I know $V_{11}=1$ but individual terms are of no consequence in the convergence of a sequence.