Verified answer

In both intervals the number of real numbers is infinite.

Let's say you have a point x in [0,1], then you want to take the distance between 0 and x (which would be x), find what it would be proportionally in [a,b] and apply that distance to be the distance between a and x', where x' is the mapped image of x. To get proportionality between [0,1] and [a,b] you have to multiply by (b-a), then shift by a to start at a.

So you get x' = (b-a)x + a

I'll let you prove that it's 1-1.

PS: you could also have a reverse map

x' = b - (b-a)x

[a,b] is like [0,1] with a magnifying glass over it so for each point

x in [0,1] we have a point in [a,b] the same percentage from start

namely , a point y such that

(y-a)/(b-a)=x giving y = (b-a)x+a

This idea will help you complete your proof.