Fourier transform example of a pure sine wave

Let's try a Fourier transform of a sine wave signal with a pure frequency, namely,
\begin{displaymath}
f(t) = \sin(2 \pi \nu t),
\end{displaymath} (1)

sampled for $t = j \Delta t$ with $j = 0, 1, \ldots{}, N-1$. To get a pure frequency, we set
\begin{displaymath}
\nu = \nu_m = m/(N \Delta t)
\end{displaymath} (2)

so that, for frequency $\nu_m$, our discretized signal becomes
\begin{displaymath}
f_j = \sin(2 \pi m j/N).
\end{displaymath} (3)