MHB Prove Inequality $(a,b,c\geq1)$

[Albert1]

$given :\,\,$ $a,b,c\geq1$

$prove:$

$(1+a)(1+b)(1+c)\geq 2(1+a+b+c)$

 

[Euge]

Here is my solution.

Spoiler

$$(1+a)(1+b)(1+c) = 1 + a + b + c + ab + bc + ca + abc $$

and since $a,b,c \ge 1$, then $ab + bc + ca \ge a + b + c $ and $abc \ge 1$; the result now follows.