According to the "Type Theory and Formal Proof" book, Church-Rosser theorem (confluence) is as follow:
Suppose that for a given term $M$, we have $M \twoheadrightarrow_\beta N_1$ and $M\twoheadrightarrow_\beta N_2$. Then there is a $\lambda-term$ $N_3$ such that $N_1\twoheadrightarrow_\beta N_3$ and $N_2\twoheadrightarrow_\beta N_3$.
But if we take $(\lambda u.v)((\lambda x.xx)(\lambda x.xx))$ expression as $M$, then it will reduce to $v$ if we start with the left sub term and reduces to $(\lambda u.v)((\lambda x.xx)(\lambda x.xx))$ if we start with the right sub term. And they would never reduce to equal terms if we keep reducing the sound one using the right sub term since it would keep converting to itself for ever.
Isn't the above example a refutation of the CR theorem, or I have missed a point somewhere?!