Proving $(1-a)(1-b)(1-c)(1-d) > 1-a-b-c-d$

Question

Prove the following inequality for $a,b,c,d \in (0,1)$: $$(1-a)(1-b)(1-c)(1-d) > 1-a-b-c-d$$

I have a problem. I don't know if my idea is good

$a=b=c=d $

$(1-a)^4 > 1- 4a $

So, this is true..

Answer 1

Hint: $$(1-a)(1-b) = 1-a-b+ab > 1-a-b.$$ Likewise, $$(1-c)(1-d) > 1-c-d.$$ Therefore, $$(1-a)(1-b)(1-c)(1-d) > [1-(a+b)][1-(c+d)]>\cdots$$