Let $\mathcal{A}$ be a non-unital commutative Banach algebra. Consider the Gelfand transform \begin{align*}\Gamma_{\mathcal{A}}:\mathcal{A}&\to C(\sigma(\mathcal{A}))\\ x&\mapsto \hat{x} \end{align*} where \begin{align*}\hat{x}:\sigma(\mathcal{A})&\to \mathbb{C}\\ h&\mapsto \hat{x}(h)=h(x) \end{align*} In Folland's A course in Abstract Harmonic Analysis it is stated that in fact $\Gamma_{\mathcal{A}}(\mathcal{A})\subset C_0(\sigma(\mathcal{A}))$, i.e. if $x\in \mathcal{A}$ then for all $\varepsilon>0$ there is a compact $K\subset \sigma(\mathcal{A})$ such that $|\hat{x}(h)|\leq \varepsilon$ for all $h\in \sigma(\mathcal{A})\setminus K$. It is not well explained why this is true.
The basic idea would be to try to get back to the unital case. We can consider the unital extension $\tilde{A}=\mathcal{A}\times \mathbb{C}$ of $\mathcal{A}$, where $\mathcal{A}\cong \mathcal{A}\times \left\{0\right\}\subset \tilde{A}$. We then have a homeomorphism (I think) \begin{align*}\sigma(\tilde{\mathcal{A}})&\cong \sigma(\mathcal{A})\cup \left\{0\right\}\\ 0&\mapsto \tilde{0}:(x,\lambda)\mapsto 0\\ \sigma(\mathcal{A})\ni h&\mapsto \tilde{h}:(x,\lambda)\mapsto h(x)+\lambda \end{align*} both are compact Hausdorff spaces in the weak*-topology induced by $\mathcal{A}^*$ and $\mathcal{A}$ respectively, while $\sigma(\mathcal{A})$ is only locally compact in general.
The above homeomorphism in turn induces a Banach algebra isomorphism $C(\sigma(\tilde{\mathcal{A}}))\cong C(\sigma(\mathcal{A})\cup\left\{0\right\})$. With this identification we have $\Gamma_{\mathcal{A}}=r\circ \left.\Gamma_{\tilde{\mathcal{A}}}\right|_{\mathcal{A}}$ where $r:C(\sigma(\mathcal{A})\cup\left\{0\right\})\to C(\sigma(\mathcal{A}))$ is the restriction map. Now I would like to use the topological/algebraic properties I know about $r$ and $\Gamma_{\tilde{\mathcal{A}}}$ to show that $\Gamma_{\mathcal{A}}(\mathcal{A})\subset C_0(\sigma(\mathcal{A}))$.
I am aware that $\mathcal{A}$ is a closed maximal ideal of $\tilde{A}$ so since $\tilde{\mathcal{A}}$ is unital, $\Gamma_{\tilde{\mathcal{A}}}(\mathcal{A})$ is itself a closed non-trivial subalgebra of $C(\sigma(\mathcal{A})\cup\left\{0\right\})$.