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