Let $f$ be a function which is continuous and L periodic. Show that your Fourier series converges in the $L^2$ metric to $f$.
Context: We are in Chapter 8 of Rudin's Principle of Mathematical Analysis. This was given as a review (ungraded) question.
Attempt: I know how to define the Fourier Series coefficients for a function that is L-periodic. I believe that they just look like this:
.
Now I believe to show that the Fourier series converges in the $L^2$ metric, I need to show that, calling the Fourier series G, for example, that: $$\int_{-l}^{l} |G-f|^2 dx<e$$ where G is the Fourier Series (at least that's what I think I am supposed to do, or something like this).
However, I honestly have no idea (after thinking about it for several hours, and looking up various things online) how to do this by applying the appropriate scaling transformation. I'm honestly not too sure even what this means (is the scaling transformation 1/L)? I found a somewhat similar proof for my question online. 
. However, this proof uses several things that I do not understand/are above the level that I am at. I would like to figure out how the Fourier series converges in the $L^2$ metric to f by applying the appropriate scaling transformation, as described in the question (and what this exactly means). Any help would be much appreciated. Thank you.