Problem in Proof of Bloch theorem

In kittel’s book on solid state physics a proof of bloch theorem is given . It says:

We consider N identical lattice points on a ring of length Na. the potential energy is periodic in a width $U(x)=U(x+sa$), where s is an integer.
Let us be guided by the symmetry of the ring to look for solutions of the wave equation such that
$$psi(x+a)=Cpsi(x)$$
where C is a constant . Then on going once around the ring
$$psi(x+Na)=psi(x)=C^Npsi(x)$$
because $psi(x)$ must be single valued. It follows that C is one of the N roots of unity or

$C=exp(i2pi s/N)$ ;
s=$0,1,2,…N-1 .$ $(1)$

We use $(1)$ to see that

$psi(x)=u_k(x)exp(i2pi sx/Na)$ $(2)$

$(2)$ is the bloch result

I couldnt follow the part where $(1)$ is used to arrive at the bloch result