Show that for all $n \in N , n \ge 1$ :

$\sum\_{k=1}^{2^n} \frac1k \ge 1 + \frac{n}2 $

Here is what I got:

I. $A(1)$: $\sum\_{k=1}^{2} \frac1k = 1.5 \ge 1.5 \;\; \implies A(1) \text{ is correct} $

II. $A(n)$ is correct: $\sum\_{k=1}^{2^n} \frac1k \ge 1 + \frac{n}2 $

III. Is $A(n+1)$ correct? $\sum\_{k=1}^{2^n+1} \frac1k \ge 1 + \frac{n+1}2 $

IV. Now I need the proof but I don't get the solution for it. Trying for hours - so hope someone can give me an answer with a reasonable explanation. Thank you very much!