I'm self-studying from the book Understanding Analysis by Stephen Abbott and I'm stuck on Theorem 1.4.5 on page 21. The aim of this theorem is to prove that $\sqrt{2}$ exists.
He starts by considering the set: $$ T = \{ t \in \mathbb{R} \mid t^2 < 2 \} $$ and set $\alpha = \mathrm{sup} \; T$. He then assumes that $\alpha^2 <2$ and writes: \begin{equation} \begin{aligned} \left( \alpha + \frac{1}{n} \right)^2 & = \alpha^2 + \frac{2 \alpha}{n} + \frac{1}{n^2} \\& < \alpha^2 + \frac{2 \alpha}{n} + \frac{1}{n} \end{aligned} \end{equation} He hasn't defined what $n$ is so I'm quite confused what the inequality exactly means. But normally, using his notation, $n \in \mathbb{N}$; if this is correct, then shouldn't the inequality sign actually be $\leq$ for the situation that $n=1$?