Prove Cauchy sequence from

Question

Can anyone prove from first principles, that $$a_n = \left(\frac{2n-1}{n}\right)_{n=1}^\infty$$ is a Cauchy sequence.

Thank you all.

I seriously suggest you to do the job yourself. I assure you there is nothing tricky involved. — Bombyx mori, Apr 17 '13 at 06:48

score 4 · Answer 1 · answered Apr 17 '13 at 06:50

4

Large HINT: Let $$a_n=\frac{2n-1}n=2-\frac1n\;.$$ Then

$$|a_n-a_m|=\left|\left(2-\frac1n\right)-\left(2-\frac1m\right)\right|=\left|\frac1m-\frac1n\right|=\frac{|m-n|}{mn}\le\frac{m+n}{mn}\;.$$

answered Apr 17 '13 at 06:50

Brian M. Scott

1 Answers1