Let $f:A\to B$ be a local homomorphism between DVR and denote with $K(A)$ and $K(B)$ the function fields of $A$ and $B$ respectively. Moreover assume that we have an embedding $i: K(A)\to K(B)$.
My question is the following:
Let $\widehat A$ and $\widehat B$ the completions with respect to the maximal ideals. Is it true that $f$ and $i$ induce a continuous embedding $\,j:K(\widehat A)\to K(\widehat B)$?
And here my solution:
any element of $K(\widehat A)$ can be expressed as $\frac{\lim a_n}{\lim b_n}$ for $a_n,b_n\in A$, so we just put:
$$j\left(\frac{\lim a_n}{\lim b_n}\right):=\frac{\lim i(f(a_n))}{\lim i(b_n)}\in K(\widehat B)$$
Is my reasoning correct? Of course we cannot apply $f$ to the denominators cause the kernel of $f$ might be non-trivial.
Many thanks in advance