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I am stuck up with these questions from my text book on sequential continuity :

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{My questions might sound trivial a bit trivial}

I am not able to figure how its being written that $|f(X_n)| \leq |X_n|$? I mean $|f(X_n)|= |X_n|$ seems fine but how come $|f(X_n)| \leq |X_n|$?

This question also writes the same but I can't figure out why?

enter image description here

From where does this inequality sign has popped up?

In need of intuitive explanation.

Thanks in advance!

Gregory Grant
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Vishal K Nair
  • 305

2 Answers2

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For the first highlight, the sine function (in absolute value) is bounded by 1 for any $x$. Similarly for the second.

Ken Wei
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Since $f(x_n)$ is 0 if $ x_n$ is a rational or is $x_n$ if $x_n$ is an irrational. In these two cases, we always have $|f(x_n)| \le |x_n|$.

Paul
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