# If $\Gamma(f):=\frac a2|f'|^2$, are there $0\le\eta_k\in C_c^\infty$ with $\eta_k\uparrow1$ and $\Gamma(\eta_k)\le1k$?

Let $$\Gamma(f):=\frac a2|f'|^2\;\;\;\text{for }f\in C_c^\infty(\mathbb R)$$ for some $$a>0$$. How can we show that there is a $$(\eta_k)_{k\in\mathbb N}\subseteq C_c^\infty(\mathbb R)$$ with $$0\le\eta_k\le\eta_{k+1}\;\;\;\text{for all }k\in\mathbb N,$$ $$\eta_k\xrightarrow{k\to\infty}1$$ and $$\Gamma(\eta_k)\le\frac1k\;\;\;\text{for all }k\in\mathbb N?$$

Yes. Let $$\eta_k\colon \mathbb{R}\to [0,1]$$ such that $$\eta_k|_{[-k,k]}=1$$, $$\eta_k|_{\mathbb{R}\setminus[-2k,2k]}=0$$ and $$\eta_k$$ is affine on $$[-2k,-k]$$ and on $$[k,2k]$$. Then the sequence $$(\eta_k)$$ is increasing to $$1$$ and $$\frac {a}2|\eta_k(x)-\eta_k(y)|^2\leq \frac{a}{2k^2}.$$ Thus $$\frac a 2|\eta_k'|^2\leq \frac 1 k$$ for a $$k$$ sufficiently large. If you want if to be smooth, simply mollify. Similar constructions work in higher dimensions (and in fact on any complete Riemannian manifold without boundary).