Prove that a sequence is not Cauchy

Question

I need help proving that the following sequence is not Cauchy: $$ \left( 1+\frac{1}{2}+\cdots+\frac{1}{n} \right)_{n=1}^\infty $$ Prove that for $m,n \ge N(\varepsilon)$ the inequality criteria holds by deriving the suitable value of $\varepsilon$.

Answer 1

let $u_n=\sum_{k=1}^{n}{\frac{1}{k}}$, let $n\geq1$

$u_{2n}-u_{n}=\sum_{k=n+1}^{2n}{\frac{1}{k}}\geq\sum_{k=n+1}^{2n}{\frac{1}{2n}}=n\frac{1}{2n}=\frac{1}{2}$.

Finally apply the Cauchy convergence definition with $\epsilon=\frac{1}{2}$