Using the finite covering theorem, it is necessary to prove that for any E>0, Y, ABS (fn (y)-f (y)).

Namely ABS (fn (y)-f (y)) < = ABS (fn (y)-fn (x))+ABS (fn (x)-f (x))+ABS (f (x)-f (y)) <; e； The second term is related to n, that is, there is N(x), when n >: when N(x), the inequality is satisfied.

According to the finite covering theorem, [a, b] is a closed interval, so there is a finite neighborhood covering [a, b]. Take the maximum value of finite N(x).