CDF of max($X$, $Y$) - where is the mistake? $X$ and $Y$ are independent r.v.'s and we know $F_X(x)$ and $F_Y(y)$. Let $Z=max(X,Y)$. Find $F_Z(z)$.
Here's my reasoning: 
$F_Z(z)=P(Z\leq z)=P(max(X,Y)\leq z)$. 
I claim that we have 2 cases here: 
1) $max(X,Y)=X$. If $X<z$, we are guaranteed that $Y<z$, so $F_Z(z)=P(Z\leq z)=P(X<z)=F_X(z)$
2) $max(X,Y)=Y$. Similarly, $F_Z(z)=P(Z\leq z)=P(Y<z)=F_Y(z)$
Since we're interested in either case #1 or #2, 
$F_Z(z)=F_X(z)+F_Y(z)-F_X(z)*F_Y(z)$
However, it's wrong and I know it. But I would like to know where the flaw in my reasoning is. I know the answer to this problem, I just want to know at what moment my reasoning fails.

$\max(X,Y)\le z$ means that both $X$ and $Y$ are $\le z$.
