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} $?