<s>[INST] Prove by mathematical induction that $1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2^n-1} > \frac{n}{2}$ 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?
 [/INST] 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.
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