Solution
= Solution
Let $a^{(m)}=(1,1/2,\ldots,1/m,0,0,\ldots)$. Each has finite support, and in the <supremum norm> metric it converges to $a=(1,1/2,1/3,\ldots)$, because $d(a^{(m)},a)=1/(m+1)$.
The limit is bounded but does not have finite support. Thus the finite-support subspace is not closed in $X$, and by part (a)(iii) \b[it is not complete]. Equivalently, the displayed <sequence> is Cauchy in that subspace but has no limit there.