# Function is pointwise limit of integrals

This is a question from some old masters exams.

Let $\phi_{n}$ be a sequence of continuous, real functions on $\mathbb{R}$ such that

$\phi_{n}(x) = 0$ for all $|x|\geq 1/n$ and $\phi_{n}(x)\geq 0$ for all $x\in\mathbb{R}$, and

$$\int\limits_{-1}^{1}{\phi_{n}(x)dx}=1$$

For each continuous function $f:\mathbb{R}\rightarrow\mathbb{R}$, let

$$f_{n}(x)=\int\limits_{-\infty}^{\infty}{\phi_{n}(x-y)f(y)dy}$$

Prove that $f_{n}(x)$ converges pointwise to $f(x)$ and prove that if $f(x)=0$ for $|x|\geq 10$, then $f_{n}$ converges uniformly to $f$.

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