Found this explanation in Manhattan strategy books 4th edition.

(1/10)^n X (l/10)^-1 < 1/10
(1/10)n X 10 < 1/10
(1/10)^n < 1/100
(1/10)^n < (1/10)^2
n>2

I fail to understand how does the inequality sign change here at the end?