 Solution

A metric space is complete when every Cauchy sequence converges to a point of the space.

Let $x^{(1)},x^{(2)},\dots$ be a Cauchy sequence in $X$, where $x^{(j)}=(x_1^{(j)},x_2^{(j)},\dots )$. For each positive integer $N$, choose $J_N$ such thatThis 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 $x=(x_1,x_2,\dots )\in X$.

For $j\ge J_N$, the first $N$ coordinates of $x^{(j)}$ agree with those of $x$, soAs $N$ is arbitrary, $x^{(j)}\to x$. Therefore