My question is simple: Is there a sequence of functions $f_n\in C_c^\infty(\mathbb R^d)$ such that
- $0\le f_n\le 1$ for all $n$
- $f_n\to 1$ pointwise
- $\int|\nabla f_n|^2\,dx\le 1/n$ for all $n$?
I know that the first two points are doable by choosing bump functions $f_n$ such that $f_n = 1$ on $B_n(0)$ and $f_n=0$ outside of $B_{n+1}(0)$.
For dimension $d=1$ I'd choose $g_n=1$ on $[-n,n]$, $g_n = 0$ on $(-\infty,-2n]\cup [2n,\infty)$ and fill up linearly in the two gaps. This function satisfies all three points, but isn't smooth. However, I'm sure $g_n$ can be approximated uniformly by a smooth function $f_n$ so that $(f_n)$ does the job. But I have no idea how to deal with higher-dimensional cases.