In my analysis course the lecturer proved that The real numbers are not countable.

I uploaded the proof however I do not understand the last part. I understand what he does, he builds nested intervals in a way that he leaves an element of R every time, however the intervals that he build, the distance converges to zero so there has to exist an x ( a unique one) in the intersection of all intervals. then he says since our sequence is equal to R , there exists an interval such that , the interval doesn't include that element which is clearly a conntradiction to the nested intervals. However what I do not understand is that what promises exactly the existence of such interval that doesn't include that unique element x.

Thanks!