Prove by mathematical induction that $1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2^n-1} > \frac{n}{2}$

Question

1)Base of induction n=2 : $${1+\frac{1}{2}+\frac{1}{3}>\frac{2}{2}}$$ 2)Assuming right for n=k $$1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2^k-1} > \frac{k}{2}$$ and now need to prove that right for n=k+1 $$1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2^k-1}+\frac{1}{2^k}+\frac{1}{2^k+1}+...+\frac{1}{2^{k+1}-1} > \frac{k+1}{2}$$ After simlifiying that we get to this $$\frac{1}{2^k}+\frac{1}{2^k+1}+...+\frac{1}{2^{k+1}-1}>\frac{1}{2}$$ any hints about further steps?

Answer 1

Hint: To show $$ \frac1{2^k}+\frac1{2^k+1}+...+\frac1{2^{k+1}-1}>\frac12 $$ count the number of terms and look at the smallest term.