So I'm trying to better understand the Discrete Fourier Transform and I have the following question. Given some integer $N$, I define the Fourier transform as a unitary matrix
$$F_N \ e_x = \frac{1}{\sqrt{N}} \sum_{y = 0}^{N - 1} e^{2 \pi i x y /N} \ e_y $$
where $e_x$ is the $x$-th element of the standard basis.
Intuitively, as $N \rightarrow \infty$ I expect $F_N$ and $F_{N - k}$ to be very close to each other (for a fixed $k$). In other words, I expect the operator $2$-norm $|| F_N - F_{N-1} ||_2$ to be bounded with something that tends to $0$ (by appropriately padding $F_{N-k}$ with identity columns). However I can't conclude a proof. The bound I arrived to have so far:
$$ || (F_{N} - F_{N-k}) e_x || \le \sum_{y = 0}^{N-k-1} \left| \frac{1}{\sqrt{N}} e^{2\pi i xy/N} - \frac{1}{\sqrt{N - k}} e^{2\pi i xy/(N - k)} \right| + \frac{m}{\sqrt{N}} $$
The second term is for the terms of the sum $\ge N - k$. Do you have any ideas/reference for this problem?