If the sum and the product of two sequences converges to zero, does that mean that each sequence converges to zero?

If the sum and the product of two sequences converges to zero, does that mean that each sequence converges to zero ?

Thanks

(Fill in the details.)

1. Prove that if $|a_n + b_n | \leq \epsilon$ and $|a_n b_n | \leq \epsilon^2$, then $|a_n - b_n| \leq \sqrt{5} \epsilon$.
   Use the fact that $(a_n - b_n)^2 = (a_n+b_n)^2 - 4 a_n b_n$.

2. Hence conclude that $|a_n| \leq \frac{1 + \sqrt{5} }{2} \epsilon$.

Let $a_n+b_n=:s_n$, $\ a_n b_n=:p_n$. Then $a_n$, $b_n$ are the two solutions of the quadratic equation $x^2-s_n x+p_n=0$. It follows that $$\max\bigl\{|a_n|,|b_n|\bigr\}={1\over2}\left(|s_n|+\sqrt{s_n^2-4p_n}\right)\to0\qquad(n\to\infty)\ .$$

$$a_n \not\to 0\quad \Longrightarrow \quad\exists\varepsilon>0, \exists n_k\ (n_k<n_{k+1}),\ |a_{n_k}| > \varepsilon \quad \Longrightarrow \quad b_{n_k} = \frac{a_{n_k}b_{n_k}}{a_{n_k}} \to 0,\ k \to \infty \quad \Longrightarrow \quad a_{n_k} = (a_{n_k}+b_{n_k}) - b_{n_k} \to 0,\ k \to \infty.$$ Contradiction.

ADDED.
Second solution. If $a_nb_n \to 0$ then $a_n\overline{b_n} \to 0$ and $\overline{a_n}b_n \to 0$. Hence $$|a_n|^2+|b_n|^2 = (a_n+b_n)(\overline{a_n+b_n}) - a_n\overline{b_n} - \overline{a_n}b_n \to 0.$$