Can the sum of all rationals under a number be finite?

Question

Does there exist any non zero number x such that the sum of all rationals between 0 and x is finite?

Answer 1

Hint: use the density of $\mathbb Q$ in $\mathbb R$ to show that for any $\varepsilon > 0$, there exists infinitely many rationals in $[\frac \varepsilon 2, \varepsilon]$, and conclude.

Answer 2

WLOG, assume $x > 0$. Note that there are infinitely many $n \in \Bbb N$ such that $\frac1n \in [0, x).$ Note that $\sum_{n \ge 1} \frac1n = \infty$. Conclude.

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Edit: Technically all I've stated above is not sufficient. You need to use the fact that there exists $N_0 \in \Bbb N$ such that $\frac1n \in [0, x)$ for all $n > N_0$.