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Let $(x_n)$ be any function sequence such that

$$ \int_0^1x_n(t)dt=1 \qquad \forall n $$

$$ \lim_{n\to\infty}x_n = x $$

I'm trying to prove that the limit $x$ also has the property $\int_0^1x(t)dt=1$. I don't think I could construct a "bounding" function to use the dominated convergence theorem. Could I have a hint?

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    Depending on the type of convergence it may or may not be true2012-11-07

3 Answers 3