It will help if you understand the scalar case before attempting the vector problem. Suppose that is a -function. Let , , , supposing that these are all finite. Taylor's theorem says that for any and s>0, for some between x and x+s. Therefore . By the triangle inequality, .
Now let . The previous inequality becomes . Square both sides and take the sup over x to get .
That proves the scalar case. Now you have to carry essentially the same argument over to the case of vector-valued mappings.