The number e as a limit of a sequence

The theorem on bounded monotone sequences has a celebrated application: it rigorously defines the number $e$.

Theorem — Euler's definition of the number $e$

The sequence $e_n = \left(1 + \frac{1}{n}\right)^n$ is increasing and bounded above. Its limit is Euler’s number $e = \lim_{n\to+\infty}\left(1+\frac{1}{n}\right)^n \approx 2{,}71828\ldots$

Since the sequence $e_n$ is monotonically increasing and bounded above, the theorem on bounded monotone sequences guarantees that the limit exists and is finite: that limit is, by definition, the number $e$.

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Topics: Sequences
Concepts: Limit of a sequence · Monotonicity · Euler’s number · Bounded sequence
Methods: The number e via a sequence
People: Leonhard Euler (Eulero)