4 10 pointslet x be the subset of l0 consisting of sequences of real n

Question

4. (10 points)Let X be the subset of l0 consisting of sequences of real numbers that have only finitely many non-zero entries: X=\left\{\left\{x_{n}\right\}_{n \geq 1} \subset \mathbf{R}: x_{n} \neq 0 \text { for only finitely many } n \geq 1\right\} \text { We equip } X \text { with the } d_{\infty} \text { metric: for two points } x=\left\{x_{n}\right\}_{n \geq 1} \text { and } y=\left\{y_{n}\right\}_{n \geq 1} \text { in } X \text {, } d_{\infty}(x, y)=\sup _{n \geq 1}\left|x_{n}-y_{n}\right| \text { (a) Show that the sequence }\left\{x^{(k)}\right\}_{k \geq 1} \subset X \text { given by } x^{(k)}=\left(1, \frac{1}{2}, \frac{1}{3}, \ldots, \frac{1}{k}, 0,0,0, \ldots\right) is a Cauchy sequence. \text { (b) Conclude that }\left(X, d_{\infty}\right) \text { is not a complete metric space. }