Original question:

Suppose $s_{n}$ is a sequence of real numbers, $l\in\mathbb{R}$ and $\displaystyle\lim_{n \to \infty}s_{n}=l.$ Prove that $\displaystyle\lim_{n \to \infty}|s_{n}|=|l|.$

I couldn't find any examples of this type of problem with the absolute values in my textbook. Is there a theorem that is related to limits with absolute value symbols? How would I construct a proof for this type of limit problem?

  • 1
    $\begingroup$ It depends what you know already. Do you know that if $f$ is any function that is continuous at $l$, then $\lim_{n\to\infty} s_n=l$ implies $\lim_{n\to\infty} f(s_n)=f(l)$? If so you can apply that fact directly. But it's also possible to do this one by hand, as Arpit Kansal's answer hints. $\endgroup$ – Greg Martin Sep 17 '16 at 18:27

Hint: If $a,b \in \mathbb R$ then $$\vert \vert a \vert - \vert b \vert \vert\leq \vert a-b \vert$$

Can you conclude from this?


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