Show $C:=\{x\in V:\|x\|\leq 1\}$ is convex

Question

Let $(V,\|\cdot\|)$ be a normed vector spcae and put $C:=\{x\in V:\|x\|\leq 1\}$. Show that $C$ is convex, which means that $0\leq t\leq 1$ $x,y\in C \Rightarrow tx+(1-t)y \in C$

My attempt:

$\|tx+(1-t)y\|\leq |t| \|x\|+|1-t| \|y\| \leq |t|+|1-t| =1$

Would this be sufficient?

Answer 1

Yes, but you should say that you assume that $x,y \in C$ and $0 \leq t \leq 1$. Other than that, this is correct.