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Problem 54: (Unions And Intersections Of Nested Sets)

Let $n$ be a natural number and suppose that $A_1,A_2,\ldots, A_n$ are sets. Suppose also that $A_1\supseteq A_2\supseteq A_3\supseteq \cdots \supseteq A_n$.

  1. Prove that $\ds\bigcup_{i=1}^n A_i = A_1$.
  2. Make a conjecture that simplifies $\ds\bigcap_{i=1}^n A_i$. Then prove your conjecture.
It's never a bad idea to start looking at a problem that involves arbitrary things by first considering specific examples. Make up some examples with 3 or 4 sets. What do you notice happening? Then make your conjecture and prove it.

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