Proof
(related to Proposition: Uniqueness of the Limit of a Sequence)
Assume, \(x\neq x'\). In the metric space \((X,d)\), set \(\epsilon=d(x,x')/2\). From the definition of convergence,:
- it follows from \(\lim_{n\rightarrow\infty} x_n=x\) that there is an \(N(\epsilon)\in\mathbb N\) with \(d(x_n,x) < \epsilon\) for all \(n > N(\epsilon)\).
- it follows from \(\lim_{n\rightarrow\infty} x_n=x'\) that there is an \(M(\epsilon)\in\mathbb N\) with \(d(x_n,x') < \epsilon\) for all \(n > M(\epsilon)\).
Therefore, for all \(n > \max(N(\epsilon),M(\epsilon))\) we have
\[d(x_n,x) < \epsilon\quad\quad\text{and}\quad\quad d(x_n,x') < \epsilon.\]
It follows that
\[d(x,x')=\le d(x,x_n) + d(x_n,x') < 2\epsilon=d(x , x'),\]
which would mean that \(d(x,x') < d(x,x')\). This is a contradiction. Therefore, the assumption \(x \neq x'\) must be wrong and both limits are identical, or \(x=x'\).
∎
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References
Bibliography
- Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983
- Forster Otto: "Analysis 2, Differentialrechnung im \(\mathbb R^n\), Gewöhnliche Differentialgleichungen", Vieweg Studium, 1984
