How to proove that a Bloch state is periodic in reciprocal space? Two Bloch states with wavevector $k$ and $k + (2\pi/a)$ are identical? I have been trying to prove that two Bloch states with wavector k and k + (2pi/a) are identical, as the book says:

can anyone help me?
I have look at some more sources in the solid-state physics, and all of them say that that the block state is periodic in reciprocal space but they never prove it.

 A: Maths reason
In order to compare the phase in two unit cells we need to pick two points where we know that $u_k$ is the same. The easiest way of doing this is to pick a point, and the same point in the next unit cell (at a distance of one lattice constant, $a$).
The phase difference from the exponential part between these locations is:
$e^{k.a}$,
if we say $k' = k + N \times 2\pi/a$
$e^{k'.a} = e^{k.a} e^{N2\pi} = e^{k.a}$,
(because $e^{N2\pi} = 1$).
IE. Adding or subtracting 2pi makes no mathematical difference, because exp(2pi) = 1, so adding 2pi to the exponent is the same as multiplying by 1.
Physics Reason
Why did we decide that we had to compare points one whole lattice constant apart again? It was because of the $u_k$, we needed it the same at each point to pick out just the effect of the exponential out the front. But that $u_k$ can include a phase gradient of its own (it generally will). What we have really shown is that we CAN (if we choose) take all of the spatial phase gradients that are steeper than $2\pi$ per unit cell and role them into the Bloch part of the wave (the $u_k$). This is using the "reduced Brillouin zone" and is the standard convention.
