Use induction to show $1+\frac14+\frac19+...+\frac1{n^2} < 2-\frac1{n}$
Assume $P(k-1)$:
$1+\frac14+\frac19+...+\frac1{(k-1)^2} < 2-\frac1{k-1}$
Show $P(k)$:
I tried to show that $2-\dfrac1{(k-1)} +\dfrac1{k^2}$ was equal to the original RHS of the equation, but that doesn't seem to be the case. Am I messing up the proof somewhere?