0
$\begingroup$

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?

$\endgroup$
  • 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
2
$\begingroup$

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?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.