Simple set theory question

[Mr Davis 97]

Homework Statement

Prove that if $A \subseteq B$, then $\bigcup A \subseteq \bigcup B$.

Homework Equations

The Attempt at a Solution

This is a simple problem, but I just want to make sure I am writing out the proof correctly:

Suppose that $A \subseteq B$. We want to show that $\bigcup A \subseteq \bigcup B$. So consider any $t \in \bigcup A$. This means that $t$ is a member of one of the sets contained in $A$. But $A \subseteq B$, so $t$ is also a member of one of the sets contained in $B$, which implies $t \in \bigcup B$.

Is there any way I could improve the proof? I just want to make sure I'm doing these right.

 

[fresh_42]

This is a simple problem, but I just want to make sure I am writing out the proof correctly.

It is.

Is there any way I could improve the proof?

No better way. You took the straight forward way and applied the definitions.
One could only mention that the double use of the set names as single sets and as a collection of sets is a bit sloppy. It would be more precise to write
$$
\forall_{\iota \in I} \quad A_\iota \subseteq B_\iota \quad \quad \Longrightarrow \quad \cup_{\iota \in I} A_\iota \subseteq \cup_{\iota \in I} B_\iota
$$
for some index set $I$. But it is clear what is meant, so the sloppiness is forgivable.

Edit: I just saw that it might be the case that $A,B$ are both meant as a collection of sets and the collection $A$ is a subcollection of $B$, i.e. every set in $A$ is also a set in $B$. In this case the notation is o.k. but it should have been mentioned.