Computing $\lim_{n \to \infty} \sup_{x \in (0,1)}\left| \frac{1}{nx} \right|$

Question

I believe that the above is equal to $\infty$ but I don't have any formal reason why.

Answer 1

Note that for all $n\in\mathbb{N}^*,$ $\sup\limits_{x\in(0,1)}\frac{1}{nx}=+\infty$ (for all $A>1,$ if you take $x=\frac{1}{A\cdot n}\in(0,1)$ you get $\frac{1}{nx}=\frac{A\cdot n}{n}=A$). Then, $$\lim\limits_{n\to+\infty}\sup\limits_{x\in(0,1)}\frac{1}{nx}=\lim\limits_{n\to+\infty}+\infty=+\infty.$$