Recall that a $\delta$-sequence can be defined as a sequence, $(\phi_n)_{n\in\mathbb N}$, of continuously differentiable, non-negative, real-valued functions for which $\int_\mathbb R\phi_ndx=1$ for all $n\in\mathbb N$. An example of a $\delta$-sequence often discussed on this website is that of, $$\phi_n(x)=\frac{n}{\sqrt\pi}e^{-n^2x^2},$$ where $n\in\mathbb N$ and $x\in\mathbb R$. Since $\phi_n(x)\to0$ as $n\to\infty$ for $|x|\to\infty$, we see that $(\phi_n)_{n\in\mathbb N}$ is a sequence of functions which vanish at infinity.
What is an example of a $\delta$-sequence which satisfies the stronger requirement of having compact support for each $n\in\mathbb N$?
Let $\phi_n(x)=\frac 1 n f(\frac x n)$ where $f(x)=c(x+1)^{2}(1-x)^{2}$ for $-1\leq x \leq 1$, with $f(x)=0$ for $x \notin [-1,1]$. Choose the positive constant $c$ such that $\int f(x)\, dx=1$.