Limit of Rational Sequence is not necessarily Rational

Theorem

Let $S = \sequence {a_n}$ be a rational sequence.

Let $S$ be convergent to a limit $L$.

Then it is not necessarily the case that $L$ is itself a rational number.

Proof

Proof by Counterexample:

By definition, Euler's number $e$ can be defined as:

$e = \ds \sum_{n \mathop = 0}^\infty \frac 1 {n!}$

Each of the terms of the sequence of partial sums is rational.

However, from Euler's Number is Irrational, $e$ itself is not.

$\blacksquare$

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.1$. Number Systems