# Proof

We have to show that the rational members $$a_n$$ of any given rational Cauchy sequence $$(a_n)_{n\in\mathbb N}$$ are bounded, i.e. there exists a positive constant $$c\in\mathbb Q$$, such that $$|a_n|\le c$$ for all $$n\in\mathbb N$$.

By definition of rational Cauchy sequences, for any rational number $$\epsilon > 0$$ we have $$|a_i-a_j| < \epsilon$$ for all $$i,j\ge N(\epsilon)$$. In particular, if we choose $$\epsilon=1$$, there is an index $$N\in\mathbb N$$ for which the absolute value of the difference $$|a_n-a_m|$$ becomes smaller than $$1$$ for all $$m,n > N$$.

Now, we consider all $$a_m$$ with the index $$m=N+1$$. Using the triangle property of the absolute value, we get

$|a_n|=|a_n-a_m+a_m|\le |a_n-a_m|+|a_m| < 1+ |a_m|\quad\quad\text{for all }n > N.$ Set $$c:=\max(|a_1|,\ldots,|a_{N}|,1+|a_m|)$$. By construction, the constant $$c$$ gives us the required result
$|a_n|\le c\quad\quad\text{for all }n\in\mathbb N.$

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### References

#### Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer VerĂ¤nderlichen", Vieweg Studium, 1983