In this answer I will treat the case in which $|\text{ }|$ is not discrete.

I first claim that $\mathfrak m_0$ is not the restriction of any proper ideal in
$k^{\infty}.$  Indeed, choose $x \in k$ such that $0 < |x| < 1$.  Then $(x^i)$
is an element of $\mathfrak m_0$ which is invertible in $k^{\infty}$ (with
inverse equal to $(x^{-i})$, and so $\mathfrak m_0$ generates the unit ideal
of $k^{\infty}$. 

This doesn't contradict anything; the maximal ideals of $k^{\infty}$ pull-back
to prime ideals in $\mathcal C(k)$ which are simply not maximal (as often happens
with maps of rings).

Furthermore, this pull-back is injective.

To see this, we first introduce some notation; namely, we
let $\mathfrak m\_{\mathcal U}$ denote the prime ideal of $k^{\infty}$ corresponding to
the non-principal ultra-filter ${\mathcal U}$,and recall that $\mathfrak m\_{\mathcal U}$ is defined as follows: an element $(x_i)$ lies in $\mathfrak m\_{\mathcal U}$ if and only
if $\{i \, | \, x_i = 0\}$ lies in in $\mathcal U$.

Now suppose that $\mathcal U_1$ and $\mathcal U_2$ are two distinct non-principal ultra-filters.  Let $A$ be a set lying in $\mathcal U_1$, but not in $\mathcal U_2$.
Then $A^c$, the complement of $A$, lies in $\mathcal U_2$.  
Choose $x \in k$ such $0 < | x | < 1,$ and let $x_i = x^i$ if $i \in A$ and
$x_i = 0$ if $i \not\in A$.  Then $(x_i)$ is an element of $\mathcal C(k)$,
in fact of $\mathfrak m_0$, and it lies in $\mathfrak m\_{\mathcal U_2}$
but not in $\mathfrak m\_{\mathcal U_1}$.

Thus $\mathfrak m\_{\mathcal U_1}$ and $\mathfrak m\_{\mathcal U_2}$ have distinct pull-backs.  

So the map 
Spec $k^{\infty} \rightarrow $ Spec $\mathcal C(k)$
is injective
and dominant (since it comes from an injective map of rings), but is not surjective.
Choosing the valuation $|\text{ }|$ allows us to add to Spec $k^{\infty}$ (which is the
Stone-Cech compactification of $\mathbb Z\_+$) an extra point dominating all the
other points at infinity (i.e. all the non-principal ultrafilters), because the valuation now gives
us a definitive way to compute limits (provided we begin with a Cauchy sequence).