[Community Question] Algebra-Precalculus: Prove that ${1\over x_1}+{1\over x_2}+\dots+{1\over x_n}\lt3$ if no $x_j=10^kx

[Community Question] Algebra-Precalculus: Prove that ${1\over x_1}+{1\over x_2}+\dots+{1\over x_n}\lt3$ if no $x_j=10^kx_i+n$ where $k,n\in\mathbb{Z^+}$

One of our user asked:

Prove that ${1\over x_1}+{1\over x_2}+\dots+{1\over x_n}\lt3$ if no $x_j=10^kx_i+n$ where $k,n\in\mathbb{Z^+}$

I have attempted this question multiple times and have barely reached anything. I tried to assume WLOG that $x_1\le x_2\le\dots\le x_n$ however I could not continue. I am still new to such inequality questions so any help would be appreciated. Thank you anyways.

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