Show that $a_n = 1 + \frac{1}{2} + \frac{1}{3} +\dotsb+ \frac{1}{n}$ is not a Cauchy sequence 
Let $$
a_n = 1 + \frac{1}{2} + \frac{1}{3} + \dotsb + \frac{1}{n} 
\quad (n \in \mathbb{N}).
$$
  Show that $a_n$ is not a Cauchy sequence even though 
  $$
\lim_{n \to \infty} a_{n+1}  - a_n = 0
$$
  (Therefore $a_n$ does not have a limit).


Hint: Showing it doesn't converge (specifically that it goes to infinity) would help.
