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Let $f:\mathbb{R}→\mathbb{R}$ be uniformly continuous; let $\{y_n\}⊂\mathbb{R}$ be such that $\lim y_n=0$; and define the sequence $f_n:\mathbb{R}→\mathbb{R}$ by $f_n(x)=f(x+y_n)$. Prove that ${f_n}$ converges uniformly to $f$.

Please explain and walk me through proving this statement.

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marked as duplicate by hardmath, ASB, Sujaan Kunalan, graydad, Najib Idrissi Mar 24 '15 at 16:16

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Let $\varepsilon > 0$. Since $f$ is uniformly continuous, there exists $\delta > 0$ such that for all $x,y \in \Bbb R$, $|x - y| < \delta$ implies $|f(x) - f(y)| < \varepsilon$. As $\lim y_n = 0$, there exists a positive integer $N$ such that $|y_n| < \delta$ for all $n \ge N$. So for $n \ge N$ and $x\in \Bbb R$, $|(x + y_n) - x| = |y_n| < \delta$, which implies $|f_n(x) - f(x)| = |f(x + y_n) - f(x)| < \varepsilon$. Since $\varepsilon$ was arbitrary, $f_n$ converges uniformly to $f$.

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