I'm studying analysis with PMA by myself and I encountered a proof that I can't understand.
In Principles of Mathematical Analysis written by Walter Rudin (page 307),
Can you explain what do the three dots (...) mean in the proof of theorem 11.10?
I'm especially curious about why the three dots are needed in the unions of sets marked by the red box.
I guess the unions of sets consist only two terms which are $A^{'}_{n} \cup A^{'}_{n-1}$ in the red box.
Then, why do we need the three dots (...) in the red box?
I understand the unions in the first parenthesis need the three dots because more than two terms are in the parenthesis.
For example, if n=3, the line becomes $A_{n}=(A^{'}_{1} \cup A^{'}_{2} \cup A^{'}_{3} ) - (A^{'}_{3} \cup A^{'}_{2})$.
If n=4, the line becomes $A_{n}=(A^{'}_{1} \cup A^{'}_{2} \cup A^{'}_{3} \cup A^{'}_{4}) - (A^{'}_{4} \cup A^{'}_{3})$.
There are only two terms in the second parentheis.
Why do we need the three dots(...) in the red box?