Pointwise convergence of 2nd degree polynomials sequence to f in [-1,1], need to prove f is a polynomial from 2nd degree at most

Question

Given a 2nd degree sequence of polynomials: $$p_n(x)=a_nx^2+b_nx+c_n$$

that pointwise converge to a function $f$ in $[-1,1]$, I need to prove that $f$ is also a polynomial from 2nd degree at most,

I used the fact that they pointwise converge to $f$ with the points $x=0,1,-1$ and proved that there exist $a,b,c\in R$ for which: $$a_n→a$$ $$b_n→b$$ $$c_n→c$$

But is it enough to prove that $f=ax^2+bx+c$ using limit arithmetics? Or should I also point out something else?

Thanks in advance

Answer 1

I'll disagree with Kavi Rama Murthy to a limited extent. Whether or not your proof is complete depends on your audience and what they understand or expect from you.

Since you yourself are unsure of whether this is sufficient, you should not just accept someone else's word for it, but instead look into why it is sufficient. In this case, it sufficient because multiplication by a constant and addition are continuous operations. That is, because

• if $u_n \to u$, then $cu_n \to cu$ for any constant $c$.
• if $u_n \to u$ and $v_n \to v$, then $u_n + v_n \to u+v$.

Since $a_n \to a, b_n \to b$, we know that $a_nx^2 \to ax^2$ and $b_nx \to bx$, since each value of $x$ or $x^2$ just plays the part of a constant multiple. And therefore $a_nx^2 + b_nx \to ax^2 + bx$. Since $c_n \to c$ as well, we finally get $$a_nx^2 + b_nx + c_n \to ax^2 + bx + c$$ That is, $p_n(x) \to ax^2 + bx + c$ for all $x$. From this it follows that $f(x) = ax^2 + bx + c$.