For example, define for a measure space $(\Omega, \mathcal{A}, \mu)$ the set
\begin{equation} A_m = \{\omega \in \Omega: |f(\omega)|\leqslant m\}, \quad m\in\mathbb{R}, \end{equation}
for measurable functions $f: \Omega \to \mathbb{\overline{R}}$, where $\mathbb{\overline{R}}$ is the extended real line. Let
\begin{equation} M = \{m \in \mathbb{R}: \mu({A_m}^c)=0\}. \end{equation}
How do we know if $t = \inf \{M\} \in M$? By the definition of the infimum, we have that for all $\varepsilon>0$, there exists $m\in M$ such that $m \leqslant t+\varepsilon$. Furthermore, since $M$ is a connected subset of $\mathbb{R}$, it must be that $t+\varepsilon \in M$ for all $\varepsilon>0$. This obviosly holds as $\varepsilon$ approaches zero, but how do I show that it also holds in the limit? I have tried defining $t_n = t+1/n$ and then letting $n \to \infty$, but I don't know where to go from there. Thanks in advance!