Question about the limit of the product of two sequences

Question

Lets suppose $a_n$ and $b_n$ denote two sequences.

Then does the following statement always hold?

$ \lim_{n \to \infty}(a_n+b_n)=a \land \lim_{n \to \infty}(a_n-b_n)=b \Rightarrow \lim_{n \to \infty}(a_n \cdot b_n)=\frac{a^2-b^2}4 $

Thanks in advance.

Answer 1

Hint: It follows from your assumptions that $\lim_{n\to\infty}a_n=\frac{a+b}2$ and that $\lim_{n\to\infty}b_n=\frac{a-b}2$.

Answer 2

Hint:

$$a_nb_n = \frac14 \left[(a_n+b_n)^2 - (a_n-b_n)^2\right] \xrightarrow{n\to\infty} \frac14 (a^2 - b^2)$$