Finite or Not

Suppose that $f \,:\,\mathbb{N} \to \mathbb{N}$ is injective. For any $N \in \mathbb{N}$ , suppose that $\{f(1),f(2),...,f(N)\} = \{a_1,a_2,...,a_N\}$ , where $a_1 < a_2 < \cdots < a_N$ . By the Rearrangement Lemma, $\sum_{n=1}^N \frac{f(n)}{n^2} \; \ge \; \sum_{n=1}^N \frac{a_n}{n^2}$ Since $a_1 < a_2 < \cdots < a_N$ , we deduce that $a_n \ge n$ for $1 \le n \le N$ , and hence $\sum_{n=1}^N \frac{f(n)}{n^2} \; \ge \; \sum_{n=1}^N \frac{a_n}{n^2} \; \ge \;\sum_{n=1}^N \frac{n}{n^2}$ and hence $\sum_{n=1}^N \frac{f(n)}{n^2} \; > \; \ln N \hspace{2cm} N \in \mathbb{N}$ which means that the infinite series $\sum_{n=1}^\infty \frac{f(n)}{n^2}$ diverges.