Prove that $\sum_{i=1}^{2^n} \frac{1}{i} \geq 1 + \frac{n}{2}$ holds for all $n$

Question

I know that:

$$\sum_{i=1}^{2^{n+1}} \frac{1}{i} = \bigg( \sum_{i=1}^{2^{n}} \frac{1}{i} \bigg) + \bigg( \sum_{i=1}^{2^{n}} \frac{1}{2^n + i} \bigg)$$

But I can't seem to establish this by induction:

$$\sum_{i=1}^{2^{n}} \frac{1}{2^n + i} \geq \frac{1}{2}$$

Answer 1

Hint: $$\frac{1}{2^n+i}\ge \frac{1}{2^{n+1}}\text{ for all }i=1,\ldots 2^n.$$

Answer 2

Compare

$$1+\frac12+\frac13+\frac14+\frac15+\frac16+\frac17+\frac18+\frac19+\frac1{10}+\frac1{11}+\cdots$$ to $$1+\frac12+\frac14+\frac14+\frac18+\frac18+\frac18+\frac18+\frac1{16}+\frac1{16}+\frac1{16}+\cdots$$