I'm reviewing past assignments and am still having trouble formulating a proof for this:
Consider the sequence $(x_n)$, where $x_n = (1,\frac{1}{2}, \frac{1}{3}, \ldots, \frac{1}{n}, 0, 0, \ldots)$. Determine whether $(x_n)$ converges in $l_1$.
It's simple to show that the coordinate-wise limit does not converge, but how can I show that the coordinate-wise limit is the only possible limit? Alternatively, I'm trying to show that $(x_n)$ is unbounded, which I think should be straightforward also, but is giving me trouble.