Without reducing generality, we assume \(a>b>c\)

\(\Rightarrow\dfrac{1}{\left[a,b\right]}=\dfrac{1}{ab}\le\dfrac{1}{2}.\dfrac{1}{3}=\dfrac{1}{6}\)

Same: \(\left\{{}\begin{matrix}\dfrac{1}{\left[b,c\right]}=\dfrac{1}{bc}\le\dfrac{1}{15}\\\dfrac{1}{\left[a,c\right]}=\dfrac{1}{ac}\le\dfrac{1}{10}\end{matrix}\right.\)

\(\Rightarrow VT\le\dfrac{1}{6}+\dfrac{1}{15}+\dfrac{1}{10}=\dfrac{1}{3}=VP\)  (The thing must prove)

Without reducing generality, we assume a>b>c

⇒1[a,b]=1ab≤12.13=16

Same: ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩1[b,c]=1bc≤1151[a,c]=1ac≤110

⇒VT≤16+115+110=13=VP

  (The thing must prove)