Prove $f_k(x)$ converges pointwise to $f$ on $\mathbb{R}$

Question

I am having some trouble proving that the following function converges pointwise. $f_{k}(x)=\left\{\begin{array}{ll}-k & \text { if } x<-k \\ x & \text { if }-k \leq x<k \\ k & \text { if } x>k\end{array}\right.$

I believe it converges pointwise to $f(x)=x$ but I am having trouble explaining rigorously why. Any help would be greatly appreciated.

(I am then looking to show that there is no set $E$ with $|\mathbb{R} \backslash E| < \infty$ such that the sequence converges uniformly on $E$ i.e. Egorov’s Theorem can fail if $\mu(X) = \infty$)

Answer 1

Fix $x\in\mathbb{R}$. Then if $k$ is large enough you have $|x|<k$ so that $f_k(x)=x$ for large enough $k$. The sequence $f_k(x)$ is then a constant sequence for large enough $k$ and converges to $x$.

Answer 2

Pick any $x \in \mathbb R$.

For all $k > |x|$, we see that $f_k(x) = x$.

The equality $\lim_{k \to \infty}f_k(x) = x$ follows by definition of pointwise limit.