Existence of a decomposition of an arbitrary rotation into three rotations about the $x,y,z$ axis respectively.

Consider the point on the unit sphere that you want to rotate to $(0,0,1)$. With the rotation about the $x$-axis you can rotate it to the $x$-$z$-plane. Then with the rotation about the $y$-axis you can rotate it to $(0,0,1)$. Then you can use the rotation about the $z$-axis to get all the other points on the unit sphere right, and thus all points.