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!
Best Answer
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.