two functions are uniformly continuous on some interval I and each is bounded on I then their product is also uniformly continuous on I .

two functions are uniformly continuous on some interval I and each is
bounded on I then their product is also uniformly continuous on I .

prove that if two real valued functions are uniformly continuous on some
interval I and each is bounded on I then their product f.g=f(x).g(x) is
also uniformly continuous on I . Is boundedness of each function on I is
necessary for the product