Indexed Family of Sets: Proof!

[shah_nosrat]

Here the question I need to prove followed by my attempt at the solution; [math] \bigcup_{\beta \in \mathcal{B}} A_{\beta} \subseteq \bigcup_{\alpha \in \mathcal{A}} A_{\alpha} [/math] and suppose

 

[math] \mathcal{B} \subseteq \mathcal{A} [/math]

 

My attempt at the solution, as follows:

 

Let [math] x \in \bigcup_{\beta \in \mathcal{B}} A_{\beta} [/math] such that for some [math] (\beta \in \mathcal{B}) [/math] we have [math] x \in A_{\beta} [/math].

 

Now, Pick [math] \beta \in \mathcal{B} [/math] , since [math] \mathcal{B} \subseteq \mathcal{A} [/math] we have [math] \beta \in \mathcal{A} [/math].

 

Hence we have [math] x \in A_{\alpha} [/math] for some [math] \alpha [/math].

 

Which follows: [math] x \in \bigcup_{\alpha \in \mathcal{A}} A_{\alpha} [/math]

 

Is the above proof correct?

[mathematic]

Looks good!

[DrRocket]

 

 

Is the above proof correct?

 

yes

[shah_nosrat]

Thank you!