Prove that $1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2^n-1}I tried using mathematical induction to prove this, but the problem I faced was that there are a lot of numbers between $\frac{1}{2^k-1}$ and $\frac{1}{2^{k+1}-1}$. Is it possible to prove this with induction or is there a better method?

$$
1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{2^n-1}=1+\left(\frac{1}{2^1}+\frac{1}{2^2-1}\right)+\left(\frac{1}{2^2}+\cdots+\frac{1}{2^3-1}\right)+\cdots+\left(\frac{1}{2^{n-1}}+\cdots+\frac{1}{2^n-1}\right)<n,
$$
since
$$
\frac{1}{2^{k-1}}+\cdots+\frac{1}{2^k-1}<2^{k-1}\cdot\frac{1}{2^{k-1}}=1.
$$
