Lemma: Decreasing Sequence of Suprema of Extended Real Numbers

Let $(a_n)_{n\in\mathbb N}$ be a real sequence and let $D_n:=\{a_k:~k\ge n\}$ be consecutive subsets of its carrier set. Consider the supremum of extended real numbers $\sup D_n$ for each $D_n$. Then, $\sup D_n\in \overline{\mathbb R}$ (are elements of the extended real numbers) for all $n\in\mathbb N$, and the sequence $(\sup D_n)_{n\in\mathbb N}$ is monotonically decreasing.

Proofs: 1

Definitions: 1

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983