If $X$ is uniform on $(0,1)$ how can I prove $2X$ is uniform on $(0,2)$? If $X$ is uniform on $(0,1)$ how can I prove $2X$ is uniform on $(0,2)$?
I am struggling with this question and similar but harder variants.
It makes a lot of sense to me intuitively but I am unfamiliar with rigorous arguments that are needed to prove it.
I honestly don't know where to begin; do I need to look at the density function of $2X$ and show it is $1/2$ on $(0,2)$ and $0$ elsewhere or perhaps I need to look at the distribution formula maybe?
Could anyone help me with this example but also try to motivate the steps so I can appreciate the technique and hopefully be able to see how I can adapt to harder but similar problems. 

You are right, there is a theorem stating that
If $U\sim U(0,1)$ then $V = (b — a)U \sim U(a,b)$ where $a < b$.
The proof is straightforward with the pdf:
$f(x) = \frac{1}{b-a}$ if $a<x<b$ and $0$ otherwise.
Thus, if we multiply the inequality by $2$ we have $2a<2x<2b$ and setting $a' =a, b'=2b$ and thus $f(2x) = \frac{1}{b'-a'}$ if $a'<2x<b'$
for $a=0$, $b=1$ and $x=u$ you get your answer as $b'=2$,$a'=0$ and $v=2u$
