Let be any well-defined function.
We want to express it as a sum of an even function and an odd function.
Let us define two other functions as follows:
Claim I: F(x) is an even function.
Proof I; Since by definition , so so that F(x) is indeed an even function.
Claim 2: G(x) is an odd function.
Proof 2: Since by definition , so so that G(x) is indeed an odd function.
Proof 3: indeed.