= Solution
A <metric space> is <complete> when every <Cauchy sequence> converges to a point of the space.
Let $x^{(1)},x^{(2)},\ldots$ be a Cauchy sequence in $X$, where $x^{(j)}=(x^{(j)}_1,x^{(j)}_2,\ldots)$. For each positive integer $N$, choose $J_N$ such that
$$
d(x^{(j)},x^{(k)})<2^{-N}
\qquad(j,k\geq J_N).
$$
This inequality says that the first $N$ coordinates of $x^{(j)}$ and $x^{(k)}$ agree. Consequently each coordinate $x_n^{(j)}$ is eventually constant. Let $x_n$ be its eventual value and put $x=(x_1,x_2,\ldots)\in X$.
For $j\geq J_N$, the first $N$ coordinates of $x^{(j)}$ agree with those of $x$, so
$$
d(x^{(j)},x)<2^{-N}.
$$
As $N$ is arbitrary, $x^{(j)}\to x$. Therefore
$$
\boxed{(X,d)\text{ is complete}}.
$$
Back to article page
