Determine the Supremum and Infimum of a sequence

Question

Given $S =\{\sqrt{n} - \sqrt{m}, n, m \in \mathbb{N}\}$. Determine the supremum and infimum of $S$.

Is it true that the infimum of $S$ is $-\infty$ when $\sqrt{n}$ is equal to $1$ (minimum) and $\sqrt{m}$ is equal to $\infty$ and its supremum is equal to $\infty$ when $n$ is maximum ($\infty$) and $m$ is minimum ($1$)? Can we just solve this by this approachment?

Answer 1

Yes the idea is just to show that

for n=1

$$\lim_{m\to+\infty} \sqrt{n} - \sqrt{m} = -\infty$$

and for m=1

$$\lim_{n\to+\infty} \sqrt{n} - \sqrt{m} = +\infty$$