Showing that a sequence is a Cauchy sequence

Question

Show that the following sequence is a Cauchy Sequence

$$\left(\frac{\cos(n)}{\sqrt{n}}\right)_{n=1}^{\infty}$$

Workings:

Suppose $\epsilon > 0$

Let N = ____ such that

$\left|\frac{\cos(n)}{\sqrt{n}} - \frac{\cos(m)}{\sqrt{m}}\right| < \epsilon$ $ $ $(\forall n.m > N)$

Let $n > m$ then

$\left|\frac{\cos(n)}{\sqrt{n}} - \frac{\cos(m)}{\sqrt{m}}\right|$

= $\frac{\cos(n)}{\sqrt{n}} - \frac{\cos(m)}{\sqrt{m}}$

$\leq \frac{\cos(n)}{\sqrt{n}}$

Want $\frac{\cos(n)}{\sqrt{n}} < \epsilon$

Now I'm not sure what to do. Any help will be appreciated.

Answer 1

Note that $|\cos(x)|\leq 1$ for every $x$ so that $$\left|\frac{\cos(n)}{\sqrt{n}}\right|\leq \frac{1}{\sqrt{n}}\to 0$$

Note This inequality also implies that $\lim_{n\to\infty}\frac{\cos(n)}{\sqrt{n}}=0$ and remember that every converging sequence in $\Bbb R$ is a Cauchy sequence.