The exercise is as follows;
Show that the sequence $$(a_n) = \left(\frac{(-1)^n}{\sqrt{n}}\right)_{n \in \Bbb N}$$ is a Cauchy Sequence.
Solution: Let $m > n.$ Since we are trying to show that $(a_n)$ is Cauchy, we must have that $$\lvert a_m - a_n\rvert < \varepsilon$$
for $m > n > N.$ We also have that $$\left|\frac{(-1)^m}{\sqrt{m}} - \frac{(-1)^n}{\sqrt{n}} \right| \leq \frac{1}{\sqrt{m}} + \frac{1}{\sqrt{n}} \leq \frac{2}{\sqrt{n}}.$$ So, given $\varepsilon > 0$, choose $N \geq \frac{2}{\sqrt{\varepsilon}}$ and then we have that $\lvert a_m - a_n\rvert < \varepsilon$ for $m > n > N$.
Is this correct? That is, is it fine to impose the condition that $m >n $ without losing generality?