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Egorov's Theorem - Counterexample in Infinite Case

I understand the overall steps to show that Egorov's theorem fails on sets with infinite measure. However I don't see how the defined sequence of measurable functions $$f_n = \chi_{[n,n+1]}x $$ converges pointwise to 0.

Using the definition from Axlers book MIRA, the sequence $f_1, f_2, \dots $ converges pointwise on $X$ to $f$ if $$lim_{n\to\infty} f_n(x) = f(x).$$

I think I am overthinking how this is supposed to work and am stuck in my head.

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Well if $x$ is any given number than $f_n(x)=0$ when $x<n+1$ so that $$f_n(x)=0$$ if $n$ is large enough.

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