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For a nonempty set $S\subset \mathbb{R}$ bounded above, $\sup S$ is its least upper bound: every $s\in S$ satisfies $s\le \sup S$, and every $u<\sup S$ fails to be an upper bound.

A sequence $x_n$ converges to $x$ if for every $\epsilon >0$ there is $N$ such that $n\ge N$ implies $|x_n-x|<\epsilon$.

The Euler theorem states that if $\gcd (a,d)=1$, thenLet $r_1,\dots ,r_{\varphi (d)}$ be the reduced residue classes modulo $d$. Multiplication by $a$ permutes them, soTheir product is a unit and may be cancelled, proving the theorem.