Induction Using Multi-Index

Question

Does anyone know how to use induction in the context of multi-indices?

I know the induction is done on the multi-index length, the main problem is how to split a multi-index of length $n+1$ into one of length $n$ and another of length $1$.

For example, I have the following problem: Prove that $|x^\alpha|\leq |x|^{|\alpha|}$ for all $\alpha\in\mathbb N^n$ and $x\in\mathbb R^n$.

The first inductive step is using $\alpha=e_j$: $|x^\alpha|=|x_j|\leq |x|=|x|^{|\alpha|}$. Now I suppose $|x^\alpha|\leq |x|^{|\alpha|}$ for every multi-index of length $n$. Now how to prove the result is true for $\alpha$ of length $n+1$?

Answer 1

Let $\alpha$ be a multi-index of length $n+1$ so that there exists a multi-index $\beta$ of length $n$ and an index $j$ with $\alpha = \beta + e_j$. Then $$x^\alpha = x^{\beta + e_j} = x^\beta x_j.$$ Now invoke the induction hypothesis: $$|x^\alpha| = |x^\beta x_j| = |x^\beta||x_j| \le |x|^{|\beta|}|x_j|.$$ Since $|x_j| \le |x|$ you get $|x^\alpha| \le |x|^{|\beta| + 1} = |x|^{|\alpha|}$.