Selected Papers on Precalculus by Tom M. Apostol

By Tom M. Apostol

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In this notation, the colon is read as “such that” and in this case the condition is being a multiple of 2. Another example of political interest, could be the set of all judges who are not judicial activists. I think you can see this last is not a very precise condition since 35 36 CHAPTER 3. SET THEORY there is no way to determine to everyone’s satisfaction whether a given judge is an activist. Also, just because something is grammatically correct does not mean it makes any sense. For example consider the following nonsense.

N ∑n n+1 1 7. For |r| < 1, find limn→∞ k=0 rk . Hint: First show k=0 rk = rr−1 − r−1 . 9. )n+1 ( )n ( 1 8. Using the binomial theorem prove that for all n ∈ N, 1 + n1 ≤ 1 + n+1 . (n) n·(n−1)···(n−k+1) Hint: Show first that k = . By the binomial theorem, k! nk . 54 CHAPTER 4. nk) ( n+1 1 1 + n+1 except you replace n with n + 1 whereever binomial expansion for this occurs. Argue the term got bigger and then note that in the binomial expan( )n+1 1 sion for 1 + n+1 , there are more terms. 9. Prove by induction that for all k ≥ 4, 2k ≤ k!

Since a ∈ AC was arbitrary, this shows every point of AC is an interior point and so AC is open. 9 Let A be a nonempty set and denote by A′ the set of limit points of A. Then A ∪ A′ is a closed set and it is the smallest closed set containing A. 7. COMPACTNESS 57 Proof: Is it the case that (A ∪ A′ ) is open? This is what needs to be shown if the given set is closed. Let p ∈ / A ∪ A′ . Then since p is neither in A nor a limit point of A, there exists B (p, r) such that B (p, r) ∩ A = ∅. Therefore, B (p, r) ∩ A′ = ∅ also.

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