I'm trying to prove the following statement (from T.Tao's Analysis 2 book):
"Let $k\geq 1$, let $I$ be a finite subset of $\mathbb{N}^k$, and $c\colon I\to\mathbb{R}$ be a function. Show that the function $P\colon\mathbb{R}^k \to \mathbb{R}$ defined by $P(x_1,\dots,x_k):=\sum_{(i_1,\dots , i_k)\in I}c(i_1,\dots ,i_k)x_1^{i_1}\cdots x_k^{i_k}$ is continuous".
Now, I've used mathematical induction on $k\geq 1$:
the base case is trivial, and follows from the fact that polynomials of one variable are continuous. Stuck on the inductive step I thought about the hint given by the book of using this previous result (Continuity of polynomials of two variables), and writing $P$ as $P(f(x_1,\dots,x_{k+1}),g(x_1,\dots,x_{k+1}))$, where $f,g\colon \mathbb{R}^{k+1}\to\mathbb{R}$ are such that $f(x_1,\dots,x_{k+1}):=x_1^{i_1}\cdots x_{k}^{i_k}$ and $g(x_1,\dots,x_{k+1}):=x_{k+1}^{i_{k+1}}$.
If I showed that $f,g$ so defined were continuous, would this work?
Is there a simpler approach to this question?
Note that the projections $$\pi_j:\quad{\mathbb R}^n\to{\mathbb R},\qquad x\mapsto x_j$$ are continuous. Furthermore sum and product ${\mathbb R}^2\to{\mathbb R}$ are continuous, hence sum and product of continuous functions are continuous, etc. The continuity of your functions $x\mapsto P(x)$ therefore follows from general principles.