Every $n$-ball is convex

Every $n$-ball is convex

I'm trying to show that every $n$-ball is convex. Let $B(a;r)$ be an
$n$-ball in $\mathbb{R}^n$ with center $a$ and radius $r$. What I need to
show is that for all $x,y \in B(a;r)$ we have $\theta x + (1-\theta)y \in
B(a;r)$ where $0 < \theta < 1$.
Since I know $||x-a||< r/2$ and $||y-a||<r/2$, I need to show that
$||\theta x + (1-\theta)y - a|| < r$. Then $||\theta x + (1-\theta)y - a||
\leq \theta ||x-y|| + ||y-a|| < \theta r + r/2$. This inequality will be
less than $r$ if $\theta \leq 1/2$.
But with $\theta > 1/2$, I'm not sure how to proceed. Could someone give
me a hint or suggest an alternative solution?