epsilon-delta proofs 6

In preceding post, we worked on exercises in which a numerical sequence approached a value. Next case is famous as $f_n$ approaching $\infty$, that is, diverging.

Prove that $f_n=1+\frac{1}{2}+\cdots+\frac{1}{n}\rightarrow \infty$, when $n\rightarrow \infty$.
$f_1=1$
$f_2=1.5$
$f_3=1.8333\cdots$
$f_4=2.08333\cdots$
$f_5=2.28333\cdots$
$\cdots \cdots$

(proof)
$$\begin{eqnarray*} f_n &=& 1+\frac{1}{2}+\left(\frac{1}{3}+\frac{1}{4}\right)+\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)+\cdots \\  &>& 1+\frac{1}{2}+\left(\frac{1}{4}+\frac{1}{4}\right)+\left(\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}\right)+\cdots \\  &=& 1+\frac{1}{2}+\left(\frac{1}{2}\right)+\left(\frac{1}{2}\right)+\cdots \\  &=& 1+\frac{m}{2}\to \infty\quad (m\to \infty) \end{eqnarray*}$$

At the first glance, may you have thought that $f_n$ would converge with any value?