= Solution
The source prints $m\ne0$ in the zero case. Since the coordinates are indexed by positive integers, that overlaps its $m=n$ case and does not define a sequence. We use the intended coordinate vectors, with $e_m^{(n)}=1$ for $m=n$ and $e_m^{(n)}=0$ for $m\ne n$.
Every $e^{(n)}$ belongs to $X$, because its entries are bounded by one and its coordinate sequence eventually vanishes. If $n\ne k$, their <supremum norm> distance is
$$
\boxed{d(e^{(n)},e^{(k)})=1}.
$$
Indeed, the difference is $1$ at coordinate $n$, $-1$ at coordinate $k$, and zero elsewhere. No subsequence is a <Cauchy sequence>, since arbitrarily late distinct terms remain distance one apart. Every <convergent sequence> is a <Cauchy sequence>, so no <convergent subsequence> exists. Therefore $\boxed{X\text{ is not sequentially compact}}$.
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