Show that for every $\displaystyle T>0$, there is a smooth function with compact support $\displaystyle \chi : \mathbb{R}\longrightarrow[0,1]$ such that $\displaystyle \chi=1$ on $\displaystyle [-T, T]$.

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