Let’s say that and are both solutions of the differential equation — and — and that they both satisfy the initial condition — — on the same interval from the existence proof above. We will show that for all by measuring the norm of their difference:
Since is a closed interval, this maximum must be attained at a point . We can calculate
but by assumption we know that , which makes this inequality impossible unless . Thus the distance between and is , and the two functions must be equal on this interval, proving uniqueness.