Can the rational numbers be arranged in a sequence? If so, consider
any such sequence of all the rational numbers. Show that every
real number is a subsequential limit of this sequence.
Since rational number is countably infinite, I see that rational numbers can be arranged in a sequence. But I'm lost how to proceed
Best Answer
Let $q_1, q_2, \ldots$ be an enumeration of the rational numbers.
Consider some real number $r$.
Suppose $r$ has an infinite decimal expansion. By considering all possible truncations of the decimal expansion, you obtain a sequence of rational numbers $q'_1, q'_2 ,\ldots$ that converges to $r$. You can use this to get a subsequence of your original sequence $(q_i)$ that converges to $r$. (Start your new sequence with $q'_1$, which must appear in the original sequence somewhere. If $q'_2$ appears before $q'_1$ in the original sequence, skip it; else append $q'_2$ to your sequence. Do the same for $q'_3$, and so on.)
Suppose $r$ has a finite decimal expansion. You can do a silly trick to give it an infinite decimal expansion (e.g. $1.2 = 1.1999\cdots$, or $-2.348 = -2.3479999\cdots$) and then perform the above procedure.