The $L^\infty$ norm of Hardy-Littlewood function equal the $L^\infty$ norm of the original function

Question

Since $$Mf(x) = \sup_{x\in I} \frac{1}{|I|} \int_I |f(y)| dy \leq \|f\|_{\infty},$$ we have $$ \|Mf\|_{\infty} \leq \|f\|_{\infty}.$$ On the other hand, it holds $$|f(x)| \leq Mf(x) \quad a.e.,$$ which means $$\|f\|_{\infty} \leq \|Mf\|_{\infty}.$$ Therefore, $\|f\|_{\infty} = \|Mf\|_{\infty}.$ Am I right?