# real analysis – divergence of harmonic series – inductive

I'm trying to show that the Harmonic series diverge through induction. So far I have shown: If we let $$s_ {n} = sum_ {k = 1} ^ {n} frac {1} {k}$$

1. $$s_ {2n} geq s_ {n} + frac {1} {2}, forall n$$
2. $$s_ {2 ^ {n}} geq 1 + frac {n} {2}, forall n$$ by induction

The next step is to derive the divergence from $$sum_ {n = 1} ^ { infty} frac {1} {n}$$, I know that it diverges, but I do not directly see how the above two parts help.

Can I only say that since then? $$s_ {2 ^ {n}} geq 1 + frac {n} {2}$$ then, since n tends to infinity, the partial sums of $$s_ {2_ {n}}$$ diverge? But that is for $$s_ {2 ^ {n}}$$ and not $$s_ {n}$$?