Okay, I definitely have to use induction I proved for the base case when $n=1$ and it was easy, but how do I prove when $n=k+1$?
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3$$\frac{1}{2^n+1} + \frac{1}{2^n+2} + \frac{1}{2^n+3} + \cdots + \frac{1}{2^{n+1}} \ge \frac{1}{2}$$ because there are $2^n$ terms in that series and each of them is at least $\frac{1}{2^{n+1}}$. – 2012ssohn Nov 10 '16 at 00:28
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See https://en.wikipedia.org/wiki/Harmonic_series_(mathematics)#Comparison_test. This is one of the oldest proofs, given by Oresme in the 14th century. – lhf Nov 10 '16 at 01:04