I need help with the following question:
Suppose $(X, \mathcal{S}, \mu)$ is a measure space with $\mu(X) < \infty$. Suppose $f_1,f_2,\dots$ is sequence of $\mathcal{S}$-measurable functions from $X$ to $\mathbb{R}$ such that $\lim\limits_{k \rightarrow \infty} f_k(x) = \infty$ for each $x \in X$. Prove that for every $\epsilon > 0$, there exists a set $E \in \mathcal{S}$ such that $\mu(X \setminus E) < \epsilon$ and $f_1,f_2,\dots$ converges uniformly to $\infty$ on $E$ (meaning that for every $t>0$, there exists $n \in \mathbb{Z}^+$ such that $f_k(x) > t$ for all integers $k \geq n$ and all $x \in E$).
I also have access to Egorov's Theorem which states the following:
Suppose $(X,\mathcal{S},\mu)$ is a measure space with $\mu(X)<\infty$. Suppose $f_1,f_2,\dots$ is a sequence of $\mathcal{S}$-measurable functions from $X$ to $\mathbb{R}$ that converges pointwise on $X$ to a function $f: X \rightarrow \mathbb{R}$. Then for every $\epsilon>0$, there exists a set $E \in \mathcal{S}$ such that $\mu(X \setminus E) < \epsilon$ and $f_1,f_2,\dots$ converges uniformly to $f$ on $E$.
I think that the above question is just a direct application of the theorem, but that convergence to infinity messes me up a little bit. Would someone be able to help, please?
HINT
Assume that $f_n \to +\infty$ pointwise.
Try mimicking Egorov's theorem proof for the sets $$A_{m,n}=\bigcup_{k=n}^{\infty}\{x:f_k(x) \leq m\}$$
https://en.wikipedia.org/wiki/Egorov%27s_theorem Here is a link to the proof,for the case of uniform convergence to a function.Mimic this proof.
Do the same ,when $f_n \to -\infty$ pointwise,using the sets $$A_{m,n}=\bigcup_{k=n}^{\infty}\{x:f_n(x)\geq-m\}$$