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**phys251** $\displaystyle "\text{Given } \{A_\alpha\}_{\alpha \in I}, J\subset I, \text{show } \bigcup_{\alpha \in J} A_\alpha \subset\bigcup_{\alpha \in I} A_\alpha."$

Here is what I did:

1. $\displaystyle \bigcup_{\alpha \in J} A_\alpha = \{x \in S: x \in A_\beta \text{ for some } \beta \in J\}, (*)$

$\displaystyle \bigcup_{\alpha \in I} A_\alpha = \{x \in S: x \in A_\beta \text{ for some } \beta \in I\}. (**)$

2. $\displaystyle y \in J \Rightarrow y \in I.$

3. $\displaystyle x \in A_\beta \text{ for some } \beta \in J \Rightarrow x \in A_\beta \text{ for some } \beta \in I.$

4. $\displaystyle \bigcup_{\alpha \in J} A_\alpha \subset \bigcup_{\alpha \in I} A_\alpha. \; \Box$

I just want to verify that this works, particularly the third step.