show directly from the definition that the following are cau

show directly from the definition that the following are cauchy sequences n/(n+1)

Solution

Given e > 0, take N = 1/e. Then for n,m > N, |(n+1)/n - (m+1)/m| = |1 + 1/n - 1 - 1/m| = |(m-n)/(mn)|. Let\'s say m>n (if not, just switch m and n). Then |(m-n)/(mn)| < |m/mn| = |1/n| < |1/(1/e)) = e as required, so the sequence is cauchy.