Proving any function can be written in the form f = O + E ?

If is any function such that:

And is any function such that:

Then prove that any given function can be expressed such that:

Now, I will show what work I've done and see if anybody can tell me if I'm on the right track. Lets assume firstly that is even. Let be any choosen odd function and be any choosen even function; then we can set:

Then we know that f can be written as:

Now, if we let:

then:

So is either even or odd, so if is even we can write it as:

Assuming is odd then we can set:

And simmiliarly to the above:

So we know g can be either even or odd. So we let g stand in for our even function, and we can write f as:

But, how do I now prove the cases where we assume and when is not even or odd? Any guidance would be appreciated. Thanks