[Math] nice open set proof that multiplication is continuous

Question

For students in a first course in analysis or topology, proving that certain function are continuous can be very tricky. However, some proofs which are difficult for students to prove using the $\epsilon-\delta$ definition of continuity are much easier to prove using the topological definition that the pre-image of every open set be open. For example, it is much easier for students to prove that $f(x)=x^2$ is continuous using open sets rather than $\epsilon-\delta$. One particularly challenging proof is showing that multiplication $\cdot \colon \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ is continuous. Is anyone aware of a slick way to prove multiplication is continuous using open sets? Any attempt I make seems to more or less be just as complex as the $\epsilon-\delta$ definition.

Answer 1

Not sure if you would be interested in this approach, but it is easy to prove with the sequence definition of continuity. Suppose $x_n \to x$ and $y_n \to y$, then $$ x_ny_n = (x + (x_n-x))(y + (y_n-y)) = xy + x(y_n-y) + y(x_n-x) + (x_n-x)(y_n-y) $$ so that by triangle inequality $$ |x_ny_n - xy| \leq |x| |y_n-y| + |y||x_n-x| + |x_n-x||y_n-y| $$ since $x_n \to x, y_n \to y$ all three terms on the right tend to zero.