TITLE: Proof of Frechet-Hoeffding Copula bounds
QUESTION [6 upvotes]: How is the lower Frechet-Hoeffding copula bound proved? 
In the bivariate case, it follows from $C(u_1,u_2)-C(u_1,v_2)-C(v_1,u_2)+C(v_1,v_2)\geq0$ by setting $(v_1,v_2)=(1,1)$.
I'm struggling to prove it in higher dimensions.

REPLY [3 votes]: Note that copulae are uniformly continuous with variation bound
$$ |C(x)-C(y)|\leq \|x-y\|_1=\sum_{i=1}^n |x_i-y_i|. $$
Using this result we get
$$1-C(x)=|C(1,1,...,1)-C(x)| \leq \sum_{i=1}^n |1-x_i|=n-\sum_{i=1}^n x_i$$ 
which can be rearranged to
$$ \sum_{i=1}^n x_i -n+1 \leq C(x)$$
as required.
The proof of the variation bound itself can be found for instance in "Probabilistic metric spaces" (1983) by Schweizer and Sklar.