Proof:

1. $F=\frac{m v^2}{r}$ (using centripetal force)

2. $F=m a$ (using Newton's 2nd law)

We can rewrite this as:

3. $a=\frac{F}{m}$

Substituting the value of $F$ from equation (1) into equation (3):

4. $a=\frac{m v^2}{r} \div m$

Simplifying:

5. $a=\frac{v^2}{r}$

This is the required expression for acceleration in circular motion.