TITLE: Convergence of sum of sequences and product of sequences implies convergence sum of sqares of sequences.
QUESTION [6 upvotes]: Prove or falsify:
If the following limits exist:
$$\lim_{n\to\infty}(a_n+b_n)$$
$$\lim_{n\to\infty}(a_n\cdot b_n)$$
Then the following limit exists:
$$\lim_{n\to\infty}({a_n}^2+{b_n}^2)$$
Solution attempt:
$$\lim_{n\to\infty}{{a_n}^2+{b_n}^2}=\lim_{n\to\infty}{{(a_n+b_n)}^2-2a_nb_n}=\lim_{n\to\infty}{a_n+b_n}\cdot\lim_{n\to\infty}{a_n+b_n}-2\lim_{n\to\infty}{a_nb_n}$$
So it converges too?

REPLY [10 votes]: Hint: $x^2+y^2 = (x+y)^2-2xy$.