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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!

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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.

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  • $\begingroup$ That is so simple! Thank you. $\endgroup$ – lj_growl Feb 26 at 20:37

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