Prove a Sequence is Cauchy

Question

Prove that sequence {(2n+1)/n} is Cauchy.

I understand the definition of a Cauchy sequence; however, I'm not sure how to find the necessary value of N to satisfy the prove.

I know that you can simply proof that the sequence is Cauchy by stating that it converges to 2. But, for this specific problem we are asked to use strictly the definition of a Cauchy sequence in writing the proof.

Thanks for the help in advance.

Answer 1

Hint: Without loss of generality, suppose $m>n$. Then $$\Bigg\vert \frac{2n+1}{n}- \frac{2m+1}{m}\Bigg\vert= \Bigg\vert \frac{m-n}{mn}\Bigg\vert\le\Bigg\vert \frac{m}{mn}\Bigg\vert=\frac{1}{n}$$

Answer 2

Assuming that you're considering this sequence as a sequence in $\mathbb{R}$, then the fact that $\{ \frac{2n+1}{n}\}_{n=1}^\infty$ is a convergent sequence (it converges to $2$) and using the fact that any convergent sequence is necessarily Cauchy.

Answer 3

There is a theorem"a sequence is convergent iff it is a Cauchy sequence" so it needs to prove that the given sequence is convergent which emphasises that it is Cauchy.$$a_n=2+\frac{1}{n}\to 2$$i.e. the sequence is convergent and hence it is Cauchy sequence