I was looking through proofs of Bezout's identity, and the various posts (1, 2, 3) that $gcd(a,b)$ is a linear combination $as + bt$, but I'm struggling to prove that if $d \in S$ where $S = \{as+bt | as+bt>0, a,b \in \Bbb N\}$ and $d|a$ and $d|b$ that $d$ is necessarily the smallest element in $S$. If not, then $$\exists d,d' \in S,\\ d|a,d|b,\\ d'|a, d'|b, \\d<'d$$ and the smallest element $d \in S$ is not $gcd(a,b)$.
There's some consequence of the proof that I suppose I'm missing, and admittedly, there are infinite choices of $s$ and $t$ which yield the same $d$ smallest element, for example, $d = as + bt \rightarrow d=as+bt+ab-ba=a(s+b)+b(t-a)$ so the choices of $s'=s+b$ and $t'=t-a$ yield $d=as'+bt'=as+bt$, but I'm interested in proving the uniqueness of an element $d\in S$ which divides both $a$ and $b$. Thank you!
