Fermats Theorem :

If  has a local maximum or minimum at , and if exists, then .

Hence, we need to prove that if has a local minimum, then .

Limit definition of derivative :

.

For limit to exist, left hand limit must be equal to the right hand limit.

Left hand limit :

Consider and .

Therefore, there must be   in order to satisfy above inequality

Right hand limit :

Consider and .

Therefore, there must be in order to satisfy above inequality.

These two conditions will satisfy simultaneously only when .

The function has a local maximum or minimum at , and if exists, then .