# Convergent subsequence for bounded rational sequences

I've seen a few different versions of the Bolzano-Weierstrass theorem in some it is specified that the sequence is real and in some it is not specified at all. Does Bolzano-Weierstrass' theorem work for rational sequences, i.e. If a sequence is rational and bounded does it then always have a convergent subsequence?

It has a subsequence which converges to a real number but not necessarily to a rational number

For example $$3, \frac{31}{10}, \frac{314}{100}, \frac{3141}{1000}, \ldots$$ is the start of a bounded rational sequence with every subsequence converging to $$\pi$$, which is not rational