In page 'Completeness of the real numbers' of Wikipedia is said that: '' completeness (of real) is equivalent to the statement that any infinite string of decimal digits is actually a decimal representation for some real number.''
I want formalize this idea: let $\mathbb{K}$ an ordered field. For simplicity i considere binary representation instead decimal representation. $\mathbb{K}$ has a natural metric structure ($d(x,y)=\lvert x-y\rvert$ as in $\mathbb{R}$) and is definited in particular the notion of series $\displaystyle\sum\limits_{n=0}^\infty \dfrac{a_n}{2^n}$ where $a_n\in\{0,1\}$ forall $n$ ($2=1+1$ where $1$ is unit of $\mathbb{K}$).
Now if $\mathbb{K}$ is complete (i.e. Least-upper-bound property), then $\mathbb{K}$ is isomorphic to $\mathbb{R}$ and over binary series converges. I ask the converse: if every binary series is convergent, then $\mathbb{K}$ is complete?
I must assume that $\mathbb{K}$ is archimedean field?