Prove that the product of the digits of the decimal representation of a natural number never exceeds the number itself.

Question

Here's my proof. Let $x$ be an arbitrary positive integer whose decimal expansion is $a_0 +\cdots+ a_n10^n$. Then, $x=a_0+ \cdots +a_n10^n \geq a_n10^n \geq a_n \cdots a_0$, since each $a_k$ is an integer in $[0,9]$.

This proof looks correct to me, but a popular resource for contest mathematics had a much longer inductive proof of this. I just want verification of my (considerably shorter) proof. Here's the resource: http://www.cheenta.com/2017/05/15/isi-entrance-2017-problem-no-5/.

Answer 1

Your proof seems correct to me