How to prove that a language $A$ is decidable?
You know that a language $L$ is decidable if there is a TM, $M$ which when given a word $w$ as input always halts and accepts if $w\in L$ or halts and rejects if $w\notin L$. Here are some hints to get you started. 

($\Rightarrow$) This is easy. If $L$ is decidable by a TM $D$, you can make an enumerator TM $E$ by trying all possible inputs in order. For word $w_i$ in that order $E$ will run $M$ on $w_i$ and write $w_i$ if $M$ accepts.

($\Leftarrow$) Suppose you had an enumerator $E$ that lists all $w\in L$ in lexicographic order. Use that to construct a decider $M$ which when given $w$ as input, runs $E$ until it finds $w$ among the enumerated strings or fails to find $w$. Your task is to show that it will always either find $w$ or decides it does not appear on the enumeration in a finite amount of time.