Proving an inequality using induction Use induction to prove the following:
$\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{2^n}\geq1+\frac{n}{2}$
What would the base case be? Would it still be $n=0$
so $\frac{1}{1}+\frac{1}{2}\geq 1+\frac{0}{2}$, which holds true.
then how would you prove for $n$ and $n+1$ to prove the proof with induction?

Base case for $n = 0$: $\frac{1}{1} \geq 1$
Inductive step: Assume $\displaystyle \sum_{k = 1}^{2^n} \frac{1}{k} \geq 1 + \frac{n}{2}$
Then, after substituting the inductive step, what we want to prove is $\displaystyle \sum_{k = 2^n + 1}^{2^{n+1}} \frac{1}{k} \geq \frac{1}{2}$
This is clear to show by a comparison:
$\displaystyle \sum_{k = 2^n + 1}^{2^{n+1}} \frac{1}{k} \geq 2^n \left( \frac{1}{2^{n+1}} \right) = \frac{1}{2}$
