Prove the sequence is Cauchy

Question

Prove with definition that it is Cauchy

$$a_n = \frac{n+3}{2n+1},$$ wheree $n$ is a natural number

I have seen other examples such as $\frac{1}{n}$ and such that show how to prove they are Cauchy but I am confused on how to choose $N$ in this case.

Answer 1

We have: For any $\epsilon > 0$, choose $N > \dfrac{10}{\epsilon}$, and if $m , n \ge N$, then: $\left|a_m-a_n\right|=\dfrac{5|m-n|}{(2m+1)(2n+1)}<\dfrac{5}{2m+1}+\dfrac{5}{2n+1}< \dfrac{5}{m}+\dfrac{5}{n} \le \dfrac{5}{N}+\dfrac{5}{N}= \dfrac{10}{N}< \epsilon$. This shows $\{a_n\}$ is a Cauchy sequence.

Answer 2

Notice that $$\frac{n+3}{2n+1} = \frac12 + \frac5{4n+2}$$Clearly the $\frac12$ doesn't affect the "cauchiness" of the series, so this is equivalent to seeing whether $\frac5{4n+2}$ is cauchy, which is a very similar proof to that of $\frac1n$.