prove that impulse is equal to the change in momentum

By definition of impulse

I = F∆t

Now again,

F = ma

a = (v-u)/ ∆t

=> F = m(v-u)/ ∆t

=> F ∆t = m(v-u)

=> I = mv – mu

But mv = final momentum (p_f )

And mu = initial momentum (p_i)

 => I = p_f – p_i

=> I = ∆p

Thus impulse is change in momentum, proved.