Translation of Convex Set in Vector Space is Convex

Theorem
Let $\Bbb F \in \set {\R, \C}$.

Let $X$ be a vector space over $\Bbb F$.

Let $C \subseteq X$ be convex.

Let $x \in X$.

Then $C + x$, the translation of $C$ by $x$, is convex.

Proof
Let $t \in \closedint 0 1$ and $u, v \in C + x$.

Then there exists $u', v' \in C$ such that:


 * $u = u' + x$

and:


 * $v = v' + x$

Then, we have:

Since $C$ is convex, we have:


 * $t u' + \paren {1 - t} v' \in C$

so:


 * $t u + \paren {1 - t} v \in C + x$

So we have that $C + x$ is convex.