Ta có: \(\left(a-1\right)\left(b-1\right)\left(c-1\right)>0\)
\(=\left(ab-a-b+1\right)\left(c-1\right)>0\)
\(=a+b+c-ab-bc-ca>0\)
\(=a+b+c-\frac{c}{ab}-\frac{a}{bc}-\frac{b}{ac}>0\)
\(\Leftrightarrow a+b+c>\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\) (Đúng)
Vậy \(\left(a-1\right)\left(b-1\right)\left(c-1\right)>0\) (Đpcm)