I'm doing a review/relearning of induction, and having some trouble with the family of problems that have to do with inequalities. Right now I'm struggling with this proof:
$$1 + \frac{1}{4} + \frac{1}{9} + ... + \frac{1}{n^2} < 2 - \frac{1}{n}$$ for all n > 1.
I evaluated P(2) as my base case, and got the truthful statement $\frac{5}{4} < \frac{3}{2}$, which holds.
My inductive hypothesis, therefore, is $1 + \frac{1}{4} + \frac{1}{9} + ... + \frac{1}{k^2} < 2 - \frac{1}{k}$.
This is where I get lost; if this were an equation with an equality, I would proceed with the inductive step $1 + \frac{1}{4} + \frac{1}{9} + ... + \frac{1}{k^2} + \frac{1}{(k+1)^2} = 2 - \frac{1}{k+1}$, and then make the substitution based on my inductive hypothesis to get the following: $2 - \frac{1}{k} + \frac{1}{(k+1)^2} = 2 - \frac{1}{k+1}$.
I'm still unclear as to how the substitution and subsequent manipulation change when you change the equality to an inequality.
