A profinite group $G$ is called finitely generated if there exist a finite 1-convergent subset $X$ (a subset X is called 1-convergent if $X \ N$ is finite for each $N \vartriangleleft _O G$) such that $G = \bar{<X>}$ (or equivalently $G = <X>N$ for each $N \vartriangleleft _O G$).
My question is how I can prove that $SL(\mathbb{Z}_p)$ and $SL(\mathbb{F}_p[|t|])$ are finite generated as profinite groups?
For $SL(\mathbb{Z}_p)$ my idea would be to use the fact that $\mathbb{Z}_p= \bar{\mathbb{Z}}$ is obviously finite generated so if I whould know that $SL(\mathbb{Z})$ is finite genereted (why?) then the surjective mophisms $\phi_i:SL(\mathbb{Z}) \to SL(\mathbb{Z/p^i\mathbb{Z}})$ whould show that $SL(\mathbb{Z})$ is dense in $SL(\mathbb{Z}_p) =\varprojlim SL(\mathbb{Z/p^i\mathbb{Z}}) $ . But why $SL(\mathbb{Z})$ finite generated?