I am trying to follow the steps of an induction proof but I do not understand how they got to the final answer.
At part (4), how did they get that inequality? What happened to the '2' in the inductive step inequality just above it? I am unsure how they went from the inductive step to part (4)
Based on the inductive hypothesis, you need to show that $$\sum_{i=1}^{k+1}\frac1{i^2}\le2-\frac1{k+1}\ .$$ You have already shown (second last line) that $$\sum_{i=1}^{k+1}\frac1{i^2}\le2-\frac1k+\frac1{(k+1)^2}\ .$$ So you now wish to prove $$2-\frac1k+\frac1{(k+1)^2}\le2-\frac1{k+1}\ ,$$ that is, $$-\frac1k+\frac1{(k+1)^2}\le-\frac1{k+1}\ .$$ The book leaves you to do this as an exercise.