Necessary and sufficient condition for $f_n$($x$) = $b_{n}x$+$c_{n}x^2$ to uniformly converge to zero

Question

Have been trying some questions on uniform and point-wise convergence of sequence of functions. Got stuck in this. I have to prove the following:-

Let ($b$$_n$) and ($c$$_n$) be sequences of real numbers then $\sum_{n=1}^\infty$ $\lvert b_n\rvert$ $\lt$ $\infty$ and $\sum_{n=1}^\infty$ $\lvert c_n\rvert$ $\lt$ $\infty$ is not a necessary and sufficient condition for the sequence of polynomials f$_n$($x$) = $b$$_n$$x$+$c$$_n$$x^2$ to converge uniformly to $0$ on the real line.

I am trying to let $b$$_n$ = $\frac{1}{n}$ and $c$$_n$ = $0$

Now f$_n$($x$) becomes $\frac{1}{n}$$x$ which converges to $0$ point-wise. How can I prove it does so uniformly?

And also is this correct?

Answer 1

Take $$b_n=c_n=\frac{1}{n^2}$$ $$\sum |b_n| \;\; and \;\; \sum |c_n|$$ are convergent.

for all real $x$, $$\lim_{n\to+\infty}f_n(x)=0$$ the sequence of functions $(f_n)$ converges in a pointwise way to zero.

But, as a polynomial function $$\sup \{|f_n(x)|\;, \; x\in \Bbb R\}=+\infty$$ since $$\lim_{x\to +\infty} \frac{1}{n^2}|x+x^2|=+\infty$$ thus, the convergence is not uniform.