Why $\mathrm{SL}_{2}(\mathbb{Z})$ is dense in $\mathrm{SL}_{2}(\widehat{\mathbb{Z}})$?

Question

When I read notes about congruence subgroup in modular forms, I was told that the surjectivity of $\mathrm{SL}_{2}(\mathbb{Z}) \rightarrow \mathrm{SL}_{2}(\mathbb{Z} / N \mathbb{Z})$ implies that $\mathrm{SL}_{2}(\mathbb{Z})$ is dense in $\mathrm{SL}_{2}(\widehat{\mathbb{Z}})$. I know $\widehat{\mathbb{Z}} \simeq \lim _{\leftarrow N} \mathbb{Z} / N Z$, and $\mathbb{Z}$ is of course dense in $\widehat{\mathbb{Z}}$. But why the surjective can show that? I try to use Chinese Reminder Theorem, but it doesn't work.