Proof
(related to Proposition: Uniqueness Of the Limit of a Sequence)
- By hypothesis, a real sequence $(x_n)_{n\in N}$ is convergent.
- Assume, the sequence converges against two values $x$ and $y.$
- By definition, this means that for every $\epsilon > 0$ there is:
- an index $N_x\in\mathbb N$ with $|x_n-x|<\frac\epsilon 2$ for all $n\ge N_x,$ and
- an index $N_y\in\mathbb N$ with $|x_n-y|<\frac\epsilon 2$ for all $n\ge N_y.$
- Now, from the triangle inequality, it follows $$\begin{align}
|x-y|&=|x-x_n+x_n-y|\nonumber\\
&\le|x-x_n|+|x_n-y|\nonumber\\
&<\frac\epsilon 2+\frac\epsilon 2\nonumber\\
&=\epsilon\nonumber\end{align}$$
for all $n\ge\max(N_x,N_y).$
- This means that $x-y=0,$ or $x=y.$
∎
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References
Bibliography
- Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983
