I wonder if anyone can provide an example of sets $ A_n \ n \in \mathbb{N}$ where $\# A_n < \infty$ but when taking the union over $n$ $\# (\bigcup_{n \in \mathbb{N}} A_n) = \infty$. Other than $A_n = \{n\}$
Example of sets $A_n \ n \in \mathbb{N}$ which are finite, but union $\cup A_n$ infinte.
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elementary-set-theory
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2This will make it: $A_n=\{n\}$. – 2017-02-12
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2How about $A_n = \{7,n\}$ – 2017-02-12
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2@Olba what is it that you're trying to do with your sets? Why isn't $A_n = \{n\}$ good enough? – 2017-02-12
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0It is good enough, I just wanted to know some not so trivial examples. @Omnomnomnom – 2017-02-12
1 Answers
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One possible example is the following:
$$A_n = \{0,1..,n\}$$
Then you have $\# A_n =n+1$, While the union over n is all the natural numbers which have infinite cardinality