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).


Starting from $e^x\ge 1+x$ and thus also $e^{-x}\ge 1-x$ one gets
$$
-\ln(1-x)\ge x \ge \ln(1+x)
$$
Now set $x=\frac1n$ to get
$$
\ln(n)-\ln(n-1)\ge\frac1n\ge\ln(n+1)-\ln(n)
$$
The left and right terms are telescoping in sums, so
$$
\ln(n)-\ln(m)\ge\sum_{k=m+1}^n\frac1k\ge\ln(n+1)-\ln(m+1)
$$
Since $\ln(n)\to\infty$, this shows that the sequence of partial sums can not be Cauchy.
