Global optimum in a convex space/set 
Recall that a set $S⊂\mathbb R^n$ is said to be convex if for any $x,y∈S$, and any $λ∈[0, 1]$, we have $λx+(1−λ)y∈S$.
  Let $f:\mathbb R^n→\mathbb R$ be a convex function and let $S⊂\mathbb R^n$ be a convex set. Let $x^∗$ be an element of $S$. Suppose that $x^∗$ is a local optimum for the problem of minimizing $f(x)$ over $S$, that is, there exists some $ε > 0$ such that $f(x^∗) \leqslant f(x)$ for all $x ∈ S$ for which $\|x−x^∗\| \leqslant ε$. Prove that $x^∗$ is globally optimal, that is, $f(x^∗) \leqslant f(x)$ for all $x ∈ S$.

I have been looking at this problem for a while now and I feel like I can kind of picture it in my head, but am having a hard time putting it into words. Can anyone help me out?

Suppose $x^*$ is not a global minimizer; there is some $x$ with $f(x) < f(x^*)$. Then every point $y$ on the line segment from $x^*$ to $x$ has $f(y) < f(x^*)$, too. Points on the line segment get arbitrarily close to $x^*$, so $x^*$ is not a local minimizer, either.
To prove the inequality $f(y) < f(x^*)$, apply convexity.
