I'm very sorry. I misread your question, I thought you were trying to show that d(a,A)=0 for a in the closure of A.
But your calculation is still not correct. The formula
\inf_{a\in A} (d(x,a) + d(a,a_0)) = \inf_{a\in A}d(x,a) + \inf_{a\in A} d(a,a_0)\text,
which you are using, does not hold...