Suppose I define sets [itex]D_n = \lbrace x \in [0,1] | [/itex] x has an n-digit long binary expansion [itex]\rbrace [/itex].(adsbygoogle = window.adsbygoogle || []).push({});

Now consider [itex]\bigcup_{n \in \mathbb{N}} D_n[/itex]. This is just the set of Dyadic rationals and therefore countable for sure.

Now for the question: is this equal to [itex]\bigcup_{n = 0}^{\infty} D_n[/itex]? Clearly we have [itex] D_1 \subset D_2 \subset ... \subset D_n [/itex] so I am tempted to think of this as [itex] \lim_{n \rightarrow \infty} D_n [/itex]. If I am allowed to take the limit, then it would seem that [itex]\bigcup_{n = 0}^{\infty} D_n = [0,1][/itex]. Where am I doing a naughty physicist mistake?

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# Limits in infinite unions of sets

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