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. 

What you say is a way, and here I give another point, but both are similar.
The distribution function $F_{2X}(x)=P(2X\le x)=P(X\le \frac{1}{2}x)=F_X(\frac{1}{2}x)$ and we can start from here.
