No. The real numbers are constructed as located Dedekind cuts, or modulated Cauchy sequences. There are no issues with this construction in constructive mathematics.
You are indeed right that you can't show that an increasing and bounded sequence of real numbers has a limit. But you can prove constructively that if a sequence of real numbers is Cauchy then it has a limit. And it is that last property which is the defining property of the real numbers (Cauchy completeness).
You are indeed right that you can't show that an increasing and bounded sequence of real numbers has a limit. But you can prove constructively that if a sequence of real numbers is Cauchy then it has a limit. And it is that last property which is the defining property of the real numbers (Cauchy completeness).