A characterization of discrete groups

The purpose of this article is to prove the following result. Let $G$ be a locally compact group, $\mathcal{A}(G)$ the Fourier algebra of $G,$ and $\mathcal{S} (G)=\{ u\in\mathcal{A}(G) :~\exists~c>~0$ such that $\parallel uv\parallel_{\mathcal{A}(G)}\leq~c\parallel v\parallel_{\infty}\hspace{0.2 cm}\forall ~ v\in \mathcal{A}(G)\}$. Then $G$ is a discrete group $\Longleftrightarrow \mathcal{S} (G)~\neq \{0\}$.