Let $a_n$ and $b_n$ be real sequences and let $f_n(x) = a_nx + b_nx^2$ be a sequence of polynomials. What should be the necessary and sufficient conditions on the sequences $a_n$ and $b_n$ so that the $f_n$ converges uniformly to $f(x) = 0$ on $\mathbb{R}$ ?
I have some conclusions: Since the convergence is unform, $\lim_{n\rightarrow \infty}f_n(1) = \lim (a_n + b_n) = 0$. Similarly, for $x = -1$, we have $\lim(-a_n + b_n) = 0$. Adding, we get $\lim{b_n} = 0$, and so we also have $\lim a_n = 0$. But these conditions are not sufficient.
The necessary and sufficient conditions for $f_n(x) = a_n x + b_n x^2$ to converge uniformly to $f(x) \equiv 0$ is that there exists $N > 0$ such that $a_n = b_n = 0$ for $n > N$. That is, the sequences must stabilize at $0$ after finitely many terms.
To see why this is the case, assume that $f_n \rightarrow 0$ uniformly. Then in particular, we can find $N > 0$ such that if $n > N$ then
$$ \sup_{x \in \mathbb{R}} |a_n x + b_n x^2| < 1. $$
However, $\sup_{x \in \mathbb{R}} |a_n x + b_n x^2| = \infty$ unless $a_n = b_n = 0$ and so we get $a_n = b_n = 0$ for all $n > N$.