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## Homework Statement

Prove that Hn converges given that:

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

## 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.

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