Proof of harmonic series - please explain the reasoning behind each step of this proof

Question

I would appreciate it if someone could explain the reasoning behind each step of this proof of the divergence of the harmonic series. I have made significant efforts to try and understand it but to no avail.

Thank you.

Answer 1

The proof is using subsequences because

If $a_1, a_2,a_3\dots$ is an increasing sequence and there exists some unbounded subsequence $a_{i_1}, a_{i_2}, a_{i_3}\dots$, then the original subsequence is unbounded.

You can prove this quite easily by:

• Take any $M\in\mathbb R$
• Because $a_{i_1}, a_{i_2}, a_{i_3}\dots$ is unbounded, there exists some $k$ such that $a_{i_k} > M$.
• Take $n=i_k$
• Then, $a_n>M$.

Therefore, for every $M\in\mathbb R$, there exists some $n\in\mathbb N$ such that $a_n>M$, and therefore the sequence $a_1,a_2,\dots$ is unbounded.

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In the particular case of your harmonic series, $s_n$ is the sum of the first $n$ elements of the series, and $i_k = 2^k$. The proof shows that $$s_1,s_2,s_4,s_8,\dots$$ is an unbounded sequence, and concludes that $$s_1,s_2,s_3,\dots$$ must also be an unbounded sequence.

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Now, is there any other step in the proof that is unclear?