In Abbott's proof for existence of $\sqrt{2}$, he takes the following steps (which completely, utterly, totally baffle me):
Choose 𝛼 as an upper bound for the set $T = \{t∈\mathbb R:t^2<2\}$ such that $𝛼 = \sup T$. He says that the assumption $𝛼^2<2$ violates our fact that 𝛼 is an upper bound. He goes on to prove this by writing:
$(𝛼+\frac 1n)^2=𝛼^2+\frac {2𝛼}{𝑛} +\frac {1}{𝑛^2}$
$2<𝛼^2+\frac {2𝛼}{𝑛}+\frac {1}{𝑛}$$=𝛼^2+\frac{2𝛼+1}{𝑛}$
Choose $n_0$such that$\frac {1}{𝑛_0}<\frac {2−𝛼^2}{2𝛼+1}$.
Which implies: $\frac {2𝛼+1}{n_0} < 2-𝛼^2$, which consequently implies:$(𝛼+ \frac {1}{n_0})^2 < 𝛼^2 + (2-𝛼^2)=2$, which contradicts the fact that 𝛼 is an upper bound for T.
I have no earthly idea what he did to get here, why he picked the fraction $\frac 1n$, or how he concluded what he concluded. This has been the first proof in the book to totally baffle me. In addition, proving that $𝛼^2 > 2$ is left as an exercise for the reader. Safe to say I have no idea how to do that one either. Could someone please break it down for me in more detail and maybe mention what theorems were used?
Edit: fixed a typo in the first line. The term $\frac {1}{n^2}$ was $\frac 1n$ by accident.