In some problem, I thought I need to prove the following subproblem :
Let $\{p_n(x)\}_{n\in\mathbb{N}}$ be a sequence of polynomials of degree 2 which converges to $p(x)=x^2+1$ point by point, i.e for every $x\in\mathbb{R}$. Assume that each $p_n(x)$ has 2 real roots, not necessarily distinct. Prove that this is a contradiction !
From this point, is there any generalization to this problem ?
If we prove that this convergence is uniform, we're almost done ! I mean, you can construct a fine tube around $x^2+1$ which protects it from all $p_n(x)$s, because $x^2+1$ never meets x-axis.
One problem here, is that norm of these sequence may not be bounded over $\mathbb{R}$.
Hint: write $p_n(x) = a_n x^2 + b_n x + c_n$. Evaluate at different points (e.g. $x=0, 1, -1$). What do you deduce for the coefficients as $n\to\infty$ ?
This generalizes readily to a sequence of degree $m$ polynomials $(p_n(x))$ converging to a fixed degree $m$ polynomial.