Limit of Rational Sequence is not necessarily Rational
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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
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