What is a metric on $A$ s.t.
a sequence $(x_n)$ is convergent in a metric space $X\iff$ there exists a continuous map $\phi:A\rightarrow X$ with $\phi(n)=x_n$ for all $n=0,1,2,...$?
What I know:
Let $d$ be the metric on $A$ that we are searching $d_X$ the one on $X$.
Being convergent to a point $x$ means that for all $\epsilon$ there is a $N$ s.t. $d_X(x,x_n)<\epsilon$ for all $n\geq N$.
Being continuous means that for all $a\in X$ and all $\epsilon$ there is a $\delta$ s.t. $d_X(x,a)<\delta$ implies $d(\phi(x),\phi(a))<\epsilon$.
How can I use these to find our metric?