Show that for every , there is a smooth function with compact support such that on .

OK, so I proved in a previous exercise that that if is locally integrable and is infinitely smooth with compact support, that the convolution is of class . With that, the hint given is to consider a convolution of two functions: which is the characteristic function on some interval (which is locally integrable) and . So we know the convolution would be of class which would take care of the smoothness aspect. I'm just not seeing how to produce ....