Let $X$ be a normed space. If the closed unit ball is complete then $X$ is a Banach space.

Question

I need to know how to prove this result.

I do not have a great deal with normed spaces, so I have been trying to carry out this proof but I am not able to do it correctly. How could I do it?

Answer 1

Any Cauchy sequence $(x_n)$ in $X$ is bounded. If $\|x_n\|<C$ for all $n$, consider $y_n=C^{-1}x_n$. Then $(y_n)$ is a Cauchy sequence contained in the unit ball. If the closed unit ball is complete it converges, and so $(x_n)$ converges in $X$.