In my textbook, they show their own proof that the Harmonic Series diverges, which I don't understand.
The harmonic series is defined as $s_n = \sum_{n=1}^\infty 1/n$. They have proved that for all $n \in \mathbb N$, $s_{2^n} \gt 1 + n/2$. And because it is monotonically increasing, it must be diverging to $\infty$ (since they claim it is not bounded above). But how can you deduce from this that it is not bounded above?
If a series is not bounded above, then for all $M \in \mathbb R$, there is an $N \in \mathbb R$ such that $s_n \gt M$ for all $n \gt N$. But the textbook hasn't picked an arbitrary real number. Instead, they chose $1 + n/2$ instead of, say, $n$.
Can someone please explain this? Thanks.
Use the given inequality, but use it the other way round. Choose some $M\in\mathbb{R}$, and let $n=2(M-1)$. Then $$s_{2^n} > 1 + \frac{n}{2} = M.$$ Another way of looking at this is to note that clearly $1+\frac{n}{2}$ is not bounded above, so given some $M$, simply choose $n$ such that $1+\frac{n}{2}>M$; then $s_{2^n}>M$.