Periodicity of the Fourier transform

If X[n] is, as above, a signal that repeats every $N$ samples, the Fourier transform of X[n] also repeats itself every $N$ units of frequency, that is,

$${\cal FT}\left \{ X[n] \right \} (k+N) = {\cal FT}\left \{ X[n] \right \} (k)$$

for all real values of $k$. This follows immediately from the definition of the Fourier transform, since the factor

$$V = \cos(-k\omega) + i\sin(-k\omega)$$

is unchanged when we add $N$ (or any multiple of $N$) to $k$.