Does every sequence of Real Numbers have a subsequence of rational numbers?

Question

Let $\{x_n\}$ be a sequence that does not converge, and let $M$ be a real number. I am thinking about how to show that there exists a subsequence of $\{x_n\}$ that converges to $M$. If I could show that there is a rational subsequence, then could I use the fact that every real number is the limit of a convergent sequence of rational numbers?

Answer 1

Not every sequence has a rational subsequence. For example, the sequence given by $$ x_n = \frac{\sqrt{2}}{n} $$ is a convergent sequence of irrrational numbers.

Answer 2

There are sequences of real numbers without a subsequence of rational numbers. Choose $x_{2n-1} = \sqrt{2}$ and $x_{2n} = \sqrt{3}$. You can't show that this has a subsequence converging to $M = 0$ because it does not.

Answer 3

Sequence:

$x_n =\sqrt {p_n}$, where $p_n$ are the primes enumerated.

Is there a rational subsequence ?