[a;b]=ab
[b;c]=bc
[c;a]=ca
\(\Rightarrow\frac{1}{\left[a;b\right]}+\frac{1}{\left[b;c\right]}+\frac{1}{\left[c;a\right]}=\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}\le\frac{1}{2.3}+\frac{1}{3.5}+\frac{1}{5.2}=\frac{1}{6}+\frac{1}{15}+\frac{1}{10}=\frac{1}{3}\)
=>đpcm