Epsilon-N limit Proof for product law for limit

Question

Let the $\lim_{n\to \infty} x_n=a$ and let $\lim_{n\to \infty} y_n=b$.

Let $\epsilon>0$ for some $n\ge N$

My notes says: $|(x_n)(y_n)-ab|=|x_n(y_n-b)+b(x_n-a)|$

Can someone show me the intermediate steps to the LHS of the equation?

Answer 1

Just distribute: Notice

$$ x_n(y_n-b) + b(x_n-a) = y_nx_n - x_nb + bx_n - ab = y_nx_n - ab $$

This is just a trick.

In order to show that $x_ny_n \to ab$ provided $x_n \to a$ and $y_n \to b$, we must show by definition that for any $\epsilon > 0$, we can find some $N > 0$ such that if $n > N$, then

$$ |x_ny_n - ab | < \epsilon $$

That is the reason why the starting point is $|x_ny_n - ab|$, because we want to bound this term.