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?

Hint for every $1\leq k\leq 2^n$
$$\frac{1}{2^n+k}\geq \frac{1}{2^{n+1}} $$
do every term in your sum is greater then $\frac{1}{2^{n+1}} $ and in the sum there is $2^n$ terms so  your sum is greater then $2^n$ times $\frac{1}{2^{n+1}} $  $\cdots$
