Globally Lipschitz continuous IVP, approximating a solution with $0<x(0)<1$.

Question

I'm trying to solve this question and I have done it but with $0 \leq x(t) \leq 1$ at the end. Using that f is Lipschitz continuous we know that the solution can't cross and diverge, but it could in theory converge together. Why is this not the case here?

Thanks