Proof
(related to Proposition: Convergent Real Sequences are Bounded)
- By hypothesis, \((a_n)_{n\in\mathbb N}\) is a convergent real sequence with \(\lim_{n\rightarrow\infty} a_n=a\).
- Thus, there is an \(N\in\mathbb N\) such that $|a_n - a| < 1\quad\quad\forall n\ge N.$
- By virtue of the triangle inequality, we get $$|a_n| = |a_n - a + a|\le |a_n - a| + |a| < 1 + |a|$$ for all $n\ge N.$
- Set \(B:=\max(|a_0|,|a_1|,\ldots,|a_{N-1}|,1 + |a|)\).
- With this constant, we have $|a_n| \le B$ for all $n\in\mathbb N.$
- Therefore, \((a_n)_{n\in\mathbb N}\) is bounded.
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References
Bibliography
- Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983
