This is Exercise 3 from p. 39 of Munkres: Analysis on Manifolds
Let $\mathbb R^\infty$ be the set of all "infinite-tuples" $x = (x_1, x_2, \ldots )$ of real numbers that end in an infinite string of $0$s. (See the exercises of § 1.)
Define an inner product on $\mathbb R^\infty$ by the rule $\langle x, y\rangle = \sum x_iy_i$. (This is a finite sum, since all but finitely many terms vanish.) Let $\|x - y\|$ be the corresponding metric on $\mathbb R^\infty$. Define $$e_i = (0, \ldots, 0, 1, 0, 0, \ldots);$$ where 1 appears in the i-th place. Then the $e_i$ form a basis for $\mathbb R^\infty$.
Let $X$ be the set of all the points $e_i$. Show that $X$ is closed, bounded, and non-compact.