A sequence $p_n(x)$ that converges for infinitely many values of $x$

Question

Let $(a_n)_{n \geq 1}, (b_n)_{n \geq 1}, (c_n)_{n \geq 1}$ be sequences of real numbers. Knowing that the sequence $$p_n(x)=(x-a_n)(x-b_n)(x-c_n)$$ converges for infinitely many values of $x$, prove that it converges for every $x \in \mathbb{R}$.

This is very similar to what happens to a polynomial when it is involved in something which happens "for infinitely many values": it actually happens for all values. Starting from this, I tried to write $$p_n(x)=x^3-(a_n+b_n+c_n)x^2+(a_nb_n+b_nc_n+c_na_n)x-a_nb_nc_n$$ But from here, I don't know anything about these $3$ sequences and I couldn't proceed further.

Answer 1

Hint.

If $p_n(x)$ and $p_n(y)$ both converge, then so does $$p_n(y)-p_n(x)\over y-x$$

Answer 2

Put $p_n(x)=x^3+A_nx^2+B_nx+C_n$.

Look at $\frac{p_n(r)-p_n(s)}{r-s}=r^2+rs+s^2+A_n(r+s)+B_n$. This one should also converge for infinitely many $r,s$.

Therefore,$A_nx+B_n$ converges for infinitely many $x$. Taking differences again we get that $A_n$ converges. Therefore $B_n$ converges, and $C_n$ converges.

If follows that $p_n(x)$ converges for all $x$.