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I'm trying to show the following.

For the sequence of functions $f_n, g_n$, all $\mathbb{R} \to \mathbb{R}$, $$ \lim_n \sup_x |f_n(x) - g_n(x) | = 0 $$ then $$ \lim_n \int_{\mathbb{R}} |f_n(x) - g_n(x)| = 0 $$

My attempt

$$ \begin{aligned} \lim_n \int_{\mathbb{R}} |f_n(x) - g_n(x)| &= \lim_n \left[ \lim_{a \to -\infty} \int_a^0 |f_n(x) - g_n(x)| + \lim_{b \to \infty}\int_0^b |f_n(x) - g_n(x)| \right] \\ &\le \lim_n \left[ \lim_{a \to -\infty} \sum_{i=1}^n M_i (x_i - x_{i-1}) + \lim_{b \to \infty} \sum_{i=1}^n M'_i (y_i - y_{i-1}) \right] \end{aligned} $$

for some partition $\{x_0, x_1, \cdots, x_n\}$ of $[a,0]$, and $\{y_0, y_1, \cdots, y_n\}$ of $[0,b]$, and $M_i, M_i'$ denotes supremum of each interval.

But I'm having trouble switching the position of $\lim$. Any help?

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This false.Take $f_n(x)=\frac 1 n$ for $0<x <n$, $0$ for other values of $x$ and $g_n(x)=0$ for all $x$.

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