Pointwise Addition on Real-Valued Functions is Associative

Theorem

Let $S$ be a set.

Let $f, g, h: S \to \R$ be real-valued functions.

Let $f + g: S \to \R$ denote the pointwise sum of $f$ and $g$.

Then:

$\left({f + g}\right) + h = f + \left({g + h}\right)$

Proof

 $\, \displaystyle \forall x \in S: \,$ $\displaystyle \left({\left({f + g}\right) + h}\right) \left({x}\right)$ $=$ $\displaystyle \left({f \left({x}\right) + g \left({x}\right)}\right) + h \left({x}\right)$ Definition of Pointwise Addition $\displaystyle$ $=$ $\displaystyle f \left({x}\right) + \left({g \left({x}\right) + h \left({x}\right)}\right)$ Real Addition is Associative $\displaystyle$ $=$ $\displaystyle \left({f + \left({g + h}\right)}\right) \left({x}\right)$ Definition of Pointwise Addition

$\blacksquare$