Sorry for the confusing title but I find this hard to formulate. Let me show with an example, I am supposed to prove that: $\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} +...+\frac{1}{n^2} \leq 2- \frac{1}{n}, \forall n\geq1$
So omitting the base case and hypothesis I come to the induction step which is:
$... + \frac{1}{(k+1)^2} \leq 2 - \frac{1}{k} + \frac{1}{(k+1)^2}$
And from my understanding of induction I have to rewrite the right hand side, to be something less than or equal to this expression:
$2-\frac{1}{k+1}$
What confuses me, do I need to show step by step how I come from the second expression to the third, or simply show that this inequality is true:
$2 - \frac{1}{k} + \frac{1}{(k+1)^2} \leq 2-\frac{1}{k+1} $
Like, will showing that the expression above is true, be sufficient formulation of the proof by induction?