Pointwise Addition on Real-Valued Functions is Associative

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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:

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


That is, pointwise addition on real-valued functions is associative.


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

$\blacksquare$