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## Homework Statement

Hey, I have another questions,

I need to find an example of a sequence of integrable functions fn:R -> R, n =1, 2, ...

such that

lim fn(x) = f(x) (as n -> ∞)

but lim ∫ |f(x)-fn(x)|dx ≠ 0 (as n -> ∞)

(with integral from - to + infinity)but lim ∫ |f(x)-fn(x)|dx ≠ 0 (as n -> ∞)

## The Attempt at a Solution

I've tried

fn = (x + x/n)

and f = x

the first conditions would be satisfied, but on the other hand,

will the limits and the integral be interchangeable? I've read that it is only permitted if the expression inside is bounded. |x/n| can't be bounded since it has an absolute sign wrapped around or would it?

Any suggestions? Thank you !!!

p.s. Would the term 'integrable' here mean a function that is reinmann integrable?