 Bounded rational sequences can converge outside the rationals

The rational numbers $q_n=\lfloor 10^n\sqrt{2}\rfloor /10^n$ lie in $[1,2]$ and obey $0\le \sqrt{2}-q_n<10^{-n}$. They form a convergent sequence in the real numbers, with limit of a sequence $\sqrt{2}$. Every subsequence has that same real limit. Because $\sqrt{2}$ is an irrational number, no subsequence converges to a point in the rational numbers with their usual metric. Boundedness alone therefore does not give the corresponding theorem in every metric space; completeness of the real numbers is doing real work here. The floor function simply truncates the decimal expansion in this example.