Divergence of the harmonic series

For $k\ge0$,
$$H_{2^{k+1}}-H_{2^k}=\sum_{j=2^k+1}^{2^{k+1}}\frac1j\ge2^k\cdot\frac1{2^{k+1}}=\frac12,$$
because the sum has $2^k$ terms, each at least $\frac1{2^{k+1}}$.

Adding these inequalities for $0,1,\dots,k-1$ and using $H_1=1$ gives
$$H_{2^k}\ge1+\frac k2.$$

The sequence $(H_n)$ is increasing and has a subsequence going to $+\infty$, so it is unbounded and $H_n\to+\infty$.