If $x \in \{ f \leq a \}$, then $f(x) \leq a \leq b$, so that $x \in \{ f \leq b \}$. Since $x \in \{ f \leq a \}$ was arbitrary, we have the inclusion $\{ f \leq a \} \subseteq \{ f \leq b \}$. $\square$