A proof of the associativity of convolutions

In the Stein's book Fourier Analysis, the proof of the associativity of convolutions is omitted.
Thus, I want to give a proof.

First, assume that $f, g, h$ are continuous. Then,
$$
\begin{aligned}
(f*g)*h(z) & = \frac{1}{2\pi} \int_{-\pi}^{\pi} \frac{1}{2\pi} \int_{-\pi}^{\pi} f(z-x-y)g(y)h(x) \textit{dydx}\\
& = \frac{1}{2\pi} \int_{-\pi}^{\pi} \frac{1}{2\pi} \int_{-\pi-x}^{\pi-x} f((z-x)-(y-x))g(y-x)h(x) \textit{dydx}\\
& = \frac{1}{2\pi} \int_{-\pi}^{\pi} \frac{1}{2\pi} \int_{-\pi}^{\pi} f(z-y)g(y-x)h(x) \textit{dydx}\\
& = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(z-y) \left[ \frac{1}{2\pi} \int_{-\pi}^{\pi} g(y-x)h(x) \textit{dx} \right] \textit{dy}\\
& = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(z-y) \left[ g*h(y) \right] \textit{dy}\\
& = f*(g*h)(z).
\end{aligned}
$$We used Fubini's Theorem and continuous assumption to interchange the order of integration.

In the book, we know that $f_k * g_k \rightarrow f*g$.
Thus, $(f_k * g_k) * h_k \rightarrow (f*g)*h$ and $f_k * (g_k * h_k) \rightarrow f*(g*h)$ and this leads to $$(f*g)*h = \lim_{k \rightarrow \infty} (f_k * g_k) * h_k = \lim_{k \rightarrow \infty} f_k * (g_k * h_k) = f*(g*h).$$