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You can label them with their lengths. I wouldn't call it an identification as it is a function from ##\mathbb{R}^n \longrightarrow \mathbb{R}## and information is lost. But in the next step, you build a limit to find the shortest of them, which is different from choosing one of them. If you'd chose one, then you will get any number or curve. Only the limit gets you the smallest or shortest. Its existence is guaranteed by completeness of ##\mathbb{R}## not by a process of choice. If we'd switch to ##\mathbb{Q}## we would still have a choice, but no limit.RockyMarciano said: