Let $x$ be an irrational number.
Let $\{a_0\}$ be the sequence of positive integers except for $a_0$ such that $x=a_0+K(1/a_n)$.
Let $a,b$ be integers such that $b>0$ and $gcd(a,b)=1$ and $|x-a/b|<1/b^2$.
Is there a counterexample of such $a,b,n$ such that $a/b\neq a_0 + K_{i=1}^n (1/a_i)$ for all $n\in\mathbb{Z}^+$?
$$(17\sqrt2-24)(17\sqrt2+24)=17^2\times2-24^2=289\times2-576=2$$ so $$0<\sqrt2-{24\over17}={2\over17(17\sqrt2+24)}\dot={1\over\sqrt217^2}$$ where $a\dot=b$ means $a$ is very close to $b$; in particular, $0<\sqrt2-{24\over17}<{1\over17^2}$. But the convergents to the continued fraction for $\sqrt2$ are $1/1,3/2,7/5,17/12,41/29,\dots$ so $24/17$ is not a convergent.