So in my textbook it says that if $f$ and $g$ are convex functions, then $h = f + g$ is also a convex function.
My question is, does this extend to a sum of more than just two convex functions. If $f_1,f_2,\dots,f_n$ are all convex functions, is the function $h = f_1 + f_2 + \dots + f_n$ also a convex function?
My attempt at a non formal proof, was that if you let $f$ and $g$ be defined as the sum of two convex functions, than surely you could extend the sum by simply letting your functions be defined as the sum of two convex functions.
Best Answer
Proof that sum of two convex function is a convex function is here Prove that the sum of convex functions is again convex.
On the other hand, if you have $n$ functions, you know that sum of first two is a convex. Then, You repeat this argument by adding $(f_1 + f_2)+f_3$ where the first bracket is a convex function. Therefore yes, this holds for any finite sum. It is an induction principle too.