# Convergence with Cauchy Sequences Proof.

Let $$a_n$$ be a sequence that satisfies $$|a_n - a_m| \leq \frac{2}{m + \sqrt{n}} \forall m,n.$$ Prove that $$a_n$$ is convergent.

I understand that the def. for Cauchy sequences is for every $$\varepsilon > 0,\exists N$$ such that for every $$m,n>N \implies |a_n - a_m| < \varepsilon$$. So, could I let $$\varepsilon = \frac{2}{m + \sqrt{n}}$$ and then say that the sequence is Cauchy and therefore convergent? Thank you!

Notice that $$|a_n - a_m| < \frac{2}{m}.$$
So take $$\varepsilon > 0$$ and N such that $$2/N < \varepsilon;$$ then, if $$n,m\ge N,$$ $$|a_n-a_m| < \varepsilon,$$
so $$a$$ is a Cauchy sequence and therefore convergent.