Showing that $\frac{1}{2^n +1} + \frac{1}{2^n +2} + \cdots + \frac{1}{2^{n+1}}\geq \frac{1}{2}$ for all $n\geq 1$ 
Show that $$\frac{1}{2^n +1} + \frac{1}{2^n +2} + \cdots + \frac{1}{2^{n+1}}\geq \frac{1}{2}$$ for all $n\geq 1$

I need this in order to complete my proof that $1 + \frac{n}{2} \leq H_{2^n}$, but I don't have any ideas. 
I know it is true at least for the first $20$ cases, but I can't prove it. Any suggestions?

$$\sum_{k=1}^{2^n} \frac{1}{2^n+k} \geq \sum_{k=1}^{2^n} \frac{1}{2^n+2^n} =\sum_{k=1}^{2^n} \frac{1}{2^{n+1}}$$
the rightmost sum equals $\frac{1}{2}$, in fact we have a strict inequality.
