I want to show formally that a simple monomial $p : \mathbb{R}^n \rightarrow \mathbb{R}$ where $p(x) = x_1\cdots x_n$ is continuous. It should be an easy problem but I can't figure out where to go from here. Maybe we can iterate on the first term somehow:
Let $x \in \mathbb{R}^n$; we want to show $p$ is continuous at $x$. That is, wts $\forall \epsilon > 0$, $\exists \delta$ such that,$$\|x - y\| < \delta \implies \|p(x) - p(y)\| < \epsilon$$We have\begin{align*} \|p(x) - p(y)\| &= |x_1\cdots x_n - y_1\cdots y_n|\\&= |\sum_{i = 0}^{n-1} x_{1}\cdots x_{i + 1}y_{i + 2}\cdots y_n - x_1\cdots x_{i}y_{i + 1}\cdots y_n|\\&\leq \sum_{i = 0}^{n-1}|x_{1}\cdots x_{i + 1}y_{i + 2}\cdots y_n - x_1\cdots x_{i}y_{i + 1}\cdots y_n|\\&= \sum_{i = 0}^{n-1}|(x_1 \cdots x_iy_{i + 2}\cdots y_n)(x_{i + 1} - y_{i + 1})|\\&\leq \sum_{i = 0}^{n-1}|x_1 \cdots x_iy_{i + 2}\cdots y_n|\cdot|x_{i + 1} - y_{i + 1}|\\\end{align*}