Proof Writing & Problem Solving · Superchallenge problem

Let $S_n = 1 + \dfrac{1}{2^3} + \dfrac{1}{3^3} + \cdots + \dfrac{1}{n^3}$. To prove $S_n < \dfrac54$ for all $n$, a write-up proves the STRONGER statement $S_n \le \dfrac54 - \dfrac{c}{n^2}$ for all $n \ge 2$, by induction with base case $n = 2$ and step $n \to n + 1$ (and checks $n = 1$ separately). There is exactly one constant $c$ for which both the base case and the inductive step (for every $n \ge 2$) go through. Find $c$.