TITLE: Prove that the sum of convex functions is again convex.
QUESTION [13 upvotes]: I have to prove that the sum of convex functions is again convex. 
I know the definition of convex function: $f(tx_1+(1-t)x_2)\leq t f(x_1)+(1-t)f(x_2)$ - this the first convex function, then I have the second one $g(tx_1+(1-t)x_2)\leq t g(x_1)+(1-t)g(x_2)$ 
What should I do next? Thank you for your help and time.

REPLY [5 votes]: Using the alternative definition of convexity can make the algebra somewhat more intuitive. 
Definition of convexity
Let $f$ be a real function defined on a real interval $I$, $f$ is convex on $I$ if and only if: 
$$ f(\alpha x + \beta y) \leq \alpha f(x) + \beta f(y)$$ 
$$ \forall x,y \in I \ : \forall \alpha, \beta \in \mathbb{R}_{\geq 0}, \ \alpha + \beta = 1 \ $$
First steps of the proof
let $h(x) = f(x) + g(x)$ where $f(x)$ and $g(x)$ are both convex functions...
I suppose we don't want to ruin the fun, but a quick google search gives a zillion versions of this proof.