<s>[INST] Arithmetic doubt while studying limits of sequences While studying limits of sequences, I came across these expressions.
$\left|\frac{-5}{n+2}\right|<\delta \iff\frac{5}{n+2}<\delta \iff n+2>\frac{1}{\delta }$
$n\in \mathbb N$
$\delta\in \mathbb R$ and $\delta >0$
I'm struggling to understand how it went from the second expression ($\frac{5}{n+2}<\delta$) to the third ($n+2>\frac{1}{\delta }$)
 [/INST] To treat that as an equal sign just makes it dead wrong.  What they mean is implies the next.   (Actually; we must assume $\delta > 0$ and that $n + 2 > 0$.  I presume those are conditions stated in the source that you omitted?  That $n\in \mathbb N$?)
(Note: these implications only work one way.)
$\left|\frac{-5}{n+2}\right|<\delta \implies\frac{5}{n+2}<\delta \implies n+2>\frac{1}{\delta }$
$|\frac {-5}{n+2}| < \delta \implies \frac {5}{n+2} < \delta$.
That should be clear $|\frac {-5}{n+2}|= \frac {5}{n+2}$ (if $n+2 > 0$).  We could make the stronger.  $|\frac {-5}{n+2}| < \delta \iff \frac {5}{n+2} < \delta$.
Now...
$\frac{5}{n+2}<\delta \implies n+2>\frac{1}{\delta }$
This is presumably the part you have trouble with.
$\frac{5}{n+2}<\delta \iff$
$\frac{5}{n+2}\frac{n+2}{\delta} < \delta \frac{n+2}{\delta}\iff$ (assuming $n+2 > 0; \delta > 0$)
$\frac 5{\delta} < n+2$
So $\left|\frac{-5}{n+2}\right|<\delta \iff\frac{5}{n+2}<\delta \iff n+2>\frac{5}{\delta }$
Now $\frac 1{\delta} < \frac 5{\delta}$ so
$\frac 1{\delta} < \frac 5{\delta}< n+2$.
So $\frac {5}{n+2} \le \delta \implies n+2 > \frac 1{\delta}$
But obviously that is only one way.  $n+2 > \frac 1{\delta} \implies (\frac 1{n+2} < \delta; \frac 1{\delta} < \frac 5{\delta}) \not \implies \frac {5}{n+2} < \delta$.
</s>