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A basic analysis question: Let $(\alpha_n)$ be a sequence of $\mathbb{R}$ such that $\alpha_n \le \alpha_{n+1} \le M$, where $M$ is some real number. How do we know that the least upper bound of the sequence exists?

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The completeness axiom guarantees that every bounded set of real numbers has a least upper bound. Choose $A=\{x\in\mathbb{R} : \alpha_n = x,\, n\in\mathbb{N}\}$. Since $A$ is bounded above (e.g. by $M$), then it has a least upper bound, which is the least upper bound of the sequence.