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**BG5965** **Problem: **Prove that $\displaystyle 1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n} > \frac{2n}{n + 1}$ where $\displaystyle n \geq 2$

**What I have so far:**

1) Prove for $\displaystyle n = 2$

$\displaystyle 1 + \frac{1}{2} > \frac{2(2)}{2 + 1}$

$\displaystyle 1\frac{1}{2} > \frac{4}{3}$

$\displaystyle 1.5 > 1.333... $ which is true.

**2) Assume for $\displaystyle k$**

$\displaystyle 1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{k} > \frac{2k}{k + 1}$ assumed true

**3) Assume for $\displaystyle k + 1$**

$\displaystyle 1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{k} + \frac{1}{k + 1} > \frac{2(k + 1)}{(k + 1)+1}$ not sure what to do

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How can I continue proving this, because this is where I get stuck on actually proving the inequality is true via induction.

Also, have I made any mistakes so far? If so, then what and how could I fix them.

Thanks for any help, BG