Let $f$ be a measurable function bounded by $M$ and supported on a set $E$. There is a sequence of simple functions $\{\varphi_n\}$ such that $|\varphi_k(x)|\leq|\varphi_{k+1}(x)|$ for all $k$ and $x$ and $\varphi_n(x)\rightarrow f(x)$ for all $x$.
In the text I am reading, the authors claim that each $\varphi_n$ in the sequence is also supported by $E$. But why is this true? Can't one of the $\varphi_n$ vanish at a given $x$, yet $f$ not vanish there as the sequence is increasing?