Prove that between every two rational numbers a/b and c/d that there is a rational number and there are an infinite number of rational numbers

Question

So the full problem is

Prove that between every two rational numbers $a/b$ and $c/d$ that:

1. There is a rational number
2. There are an infinite number of rational numbers

I am having some trouble on which approach I should use, but I tried to solve part 1.

Part 1) So I believe I proved that between every two rational numbers $a/b$ and $c/d$ that there is a rational number

I have my base case as $a,b,c,d$ all being integers.

My solution:

For some number $x$, $$ x = \frac{\dfrac a b + \dfrac c d}{2} = \frac{ \dfrac{ad}{bd} + \dfrac{cb}{bd}} 2 = \frac{ \dfrac{ad + cb}{bd} } 2 = \frac{ad + cb}{2bd} $$

I concluded that ad + cb must be an integers because addition/subtraction of integers leads to an integer. Same case for $2db$.

Not sure if this is correct though

Part 2) Prove that there are an infinite number of rational numbers

I am confused onto how to solve this part, where to start, and how to use the answer from part 1 into this problem.

I've been toiling over this, so your input is greatly appreciated. :)

Answer 1

Hint, between one of the original rationals and the one you just found you can find another rational number, and between one of the original rationals and the one you just found you can find another rational number...

Think induction.