Let $R$ be a random variable with a $U(0,1)$ distribution, $k$ be a nonzero integer constant, and $c$ be a real constant. Let $$S\equiv kR + c \ \mod 1$, \ \text{with} \ 0\leq S \leq 1$$ Show that $S$ has a $U(0,1)$ distribution.
Attempted proof - Let $R\sim U(0,1)$ and consider $c = 0$. So we have $S = kR$. So we have $$f_{kR}(t) = \begin{cases} 1 \ &\text{if} \ t\in (0,1)\\ 0 \ &\text{otherwise} \end{cases}$$ I am not sure if this works, I am being thrown off by the constant $k$ any suggestions on solving this problem are greatly appreciated.
Basic approach. Suppose $R \sim U(0, 1)$ and $k = 2$. Then $S = 2R \bmod 1$, with $0 \leq 2R \leq 2$.
We then consider the CDF $F_S(t) \equiv P(S \leq t)$, for $0 \leq t \leq 1$. Based on the above, we can write
\begin{align} F_S(t) & = P(0 \leq S \leq t) \\ & = P(0 \leq 2R \leq t \textbf{ or } 1 \leq 2R \leq 1+t) \\ & = P(0 \leq 2R \leq t) + P(1 \leq 2R \leq 1+t) \\ & = P\left(0 \leq R \leq \frac{t}{2}\right) + P\left(\frac{1}{2} \leq R \leq \frac{1+t}{2}\right) \\ & = F_R\left(\frac{t}{2}\right)-F_R(0) + F_R\left(\frac{1+t}{2}\right)-F_R\left(\frac{1}{2}\right) \\ & = \frac{t}{2}-0+\frac{1+t}{2}-\frac{1}{2} \\ & = \frac{t}{2}+\frac{t}{2} = t \end{align}
This establishes that $S$ is uniform on $(0, 1)$ for $k = 2$ and $c = 0$. Now generalize.