Proving the divergence of a Harmonic Series

1. The problem statement, all variables and given/known data

Prove that Hn converges given that:

[tex]H_{n}=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}[/tex]




3. The attempt at a solution
First I supposed that the series converges to H:

[tex]H_{n}=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}\geq1+\frac{1}{2}+\frac{1}{4}+\frac{1}{4}+\frac{1}{6}+\frac{1}{6}+\frac{1}{8}+\frac{1}{8}+...+\frac{1}{n}[/tex]

Which implies that [tex]H_{n}=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}\geq1+H_{n}[/tex]

Which contradicts with the series converging. Hence the series doesn't converge. I know there is another proof where you set H to be greater than an infinite sum of 1/2's but I wanted to think of something else. Is this correct? Any help would be appreciated.

 

Yes that's the proof i'm talking about because the first bracketed terms are greater than 1/4+1/4=1/2
and the next 4 terms are greater than 1/8+1/8+1/8+1/8 and so on and so forth and at the end you have a sum that's greater than an infinite sum of 1/2. That's it but I was trying to see if the proof I made up was on the right path.

 

Or how about if I restate the given information and say prove that the harmonic series:

1+1/2+1/3+1/4+... diverges.

Then now i can suppose that the harmonic series converges to H and we let:

H=1+ 1/2+ 1/3+ 1/4+ 1/5+ 1/6+ 1/7+ 1/8+...≥1+ 1/2+ 1/4+ 1/4+ 1/6+ 1/6+ 1/8+ 1/8+...

which equals 1+1/2+1/2+1/3+1/4+... = 1/2+H .

And the inequality holds because 1/3+ 1/4≥ 1/4+ 1/4 and 1/5 + 1/6 ≥1/6+1/6 and so on and so forth.

So we supposed that the harmonic series converges to H yet we found that H≥1/2 + H , which is a contradiction so therefore the series diverges. I think this is set up much better.