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I've seen the inequality

$$|f(x)-f(y)|\leq|f(x)-f_n(x)|+|f_n(x)-f_n(y)|+|f_n(y)-f(y)|$$

used many times to prove that if a sequence of continuous functions converges uniformly to $f$, then $f$ is continuous, but I never understood where it came from.

Could someone enlighten me?

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$$\begin{align} |f(x)-f(y)| &= |f(x)-f_n(x)+f_n(x)-f_n(y)+f_n(y) - f(y)| \\ &\leq |f(x)-f_n(x)|+|f_n(x)-f_n(y)|+|f_n(y)-f(y)| \end{align}$$ using the triangle inequality.

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  • $\begingroup$ Thanks a lot! I figured I had to use the triangle inequality, but I couldn't think of your clever way. $\endgroup$ – YoTengoUnLCD Dec 19 '15 at 2:15
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    $\begingroup$ I will accept your answer when it's available (also, I've never seen an answer get so many upvotes in so little time haha). $\endgroup$ – YoTengoUnLCD Dec 19 '15 at 2:16
  • $\begingroup$ @YoTengoUnLCD Haha, I've been here for 2.5 years and I'd have to say the same $\endgroup$ – Clarinetist Dec 19 '15 at 2:20

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