Copulas: Understanding U & V-P(U≤u)=C(u,1)=u?

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iomtt6076
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I'd appreciate it if someone could help clear up something I'm not understanding in a textbook I'm studying:

Given a copula C(u,v), we have [tex]P(U\leq u)=C(u,1)=u[/tex].

But why isn't it [tex]P(U\leq u)=C(u,\infty)=u[/tex]? Isn't it true that [tex]C_U(u)=P(U\leq u)=\lim_{v\to\infty}C(u,v)[/tex]?
 

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iomtt6076 said:
But why isn't it [tex]P(U\leq u)=C(u,\infty)=u[/tex]? Isn't it true that [tex]C_U(u)=P(U\leq u)=\lim_{v\to\infty}C(u,v)[/tex]?

Yes it is, since copulas are also joint distribution functions.
 
Okay, thanks; I guess the textbook should have explicitly said copulas were defined on [0,1]x[0,1].