Is the function $f(x)=1/|x|$ bounded in the interval $(-\infty,c)$, if $c<0$?

Question

Is the function $f(x)=1/|x|$ bounded in the interval $(-\infty,c)$, if $c<0$? I have to show that the function $f(x)$ is bounded in the interval $(-\infty,c)$, $c<0$, by a certain constant $K$ that depends on the $c$ chosen.

Answer 1

Yes, it is bounded. Since $x<c$ we have $|x|=-x>-c=|c|$. Hence $f(x)=1/|x|<1/|c|$ and thus the function is bounded by the constant $K=1/|c|$ on the interval $(-\infty,c)$. (and $f(x)$ is bounded from below by $f(x)\geq 0$).