I just read a Corollary in a finance course note:
Suppose the market is arbitrage free and $C$ is a contingent claim. Then $C$ is attainable if and only if it admits a unique arbitrage-free price.
I don't know how to prove the converse. Assuming we are just dealing with one period model. What I got so far: if it admits a unique arbitrage-free price, then price of a contingent claim $C$, $\Pi(C) = E_Q[\frac{C}{1+r}]$ is unique. But I am not sure how to prove $C = h \cdot S_1$ $ \textbf{P}$- $ a.s.$ for some strategies $h \in$ $\textbf{R}^{N+1}$ where there are $N$ stocks and one bond, to conclude that $C$ is attainable.