About an inequality in the proof of maximal function In the proof of the fact that if $f\in L^p$, where $1<p\leq L^p$, then $Mf\in L^p$ and $$||Mf||_p\leq A_p ||f||_p$$ ($Mf$ is maximal function)  Stein says $$|f(x)|\leq |f_1(x)|+\alpha /2 (*)$$ and from this we get $$M(f)(x)\leq M(f_1)(x)+\alpha /2(**)$$.
QUESTION: How we get (**) from (*)?
EDIT: $f_1(x)=f(x)$ if $|f(x)|\geq \alpha /2$ and $f_1(x)=0$ in other case.

The following three properties of the maximal function follow immediately from the definition:

*

*$|f| \leq |g|$ implie $Mf \leq Mg$.


*$M(f+g) \leq Mf+Mg$


*$M(c)=|c|$ if $c$ is a  constant.
Just combine these to get (**)
