TITLE: Showing Unit sphere is convex
QUESTION [6 upvotes]: Good evening guys!
I have to show that the unit sphere represented by  is convex.
A set is said to be convex when $sx + (1 - s)y \in M$, where $x, y \in M$ and $s \in (0,1)$
I've read on wikipedia that this can be proven over the triangle inequality, but I think it can be solved in another way? Would this be enough as proof:
For the unit sphere, we have to prove that $0 \leq sx + (1 - s)y \leq 1$ (because $||x||\leq 1$ therefore, $0 \leq x,y \leq 1$). Seeing as the maximal value that x and y can take are 1, the maximum the equation can achieve is 1 (when s=1,x=1 or s=0,y=1). The same can be shown for the minimum 0, therefore it is really between 0 and 1. Finished?
Many thanks in advance!

REPLY [10 votes]: Unfortunately you can not say from a vector whether it is between two reals or not. That's why you need to consider the norm of $sx+(1-s)y$. But as Wikipedia suggests, and basicly what you figured out in a different way, you see that if $x,y\in \bar{B_{X}}$ and $s\in[0,1]$, then by triangle inequality:
\begin{align*}
||sx+(1-s)y||\leq ||sx||+||(1-s)y||=s||x||+(1-s)||y||\leq s+(1-s)=1
\end{align*}
Hence $sx+(1-s)y\in \bar{B_{X}}$ and thus $\bar{B_{X}}$ is convex.