1 Infinite Sums And Infinite Products Of Cardinals Are Defined As Follows Let D Be Cardinals And Pick D With O D 1 (146.45 KiB) Viewed 50 times
I need the solution to #3 - I boxed in red. PLEASE PRINT thesolution so I can CLEARLY read the numbers and symbols.
1.* Infinite sums and infinite products of cardinals are defined as follows: Let d; be cardinals, and pick D, with o(D;) = d;. (For Ed, pick the Dis disjoint.) Then Ed; is the cardinal of U D₁, and II d; is the cardinal of the Cartesian product of the Di's. Let di, e; be cardinals with d; < e, for all i. Prove that Ed; < Ile;. (Observe that if every d; = 1 and every e; = 2 we get Theorem 6.) 2.* Prove that N > N. (Hint: By Exercise 1 we have N₁... No + N + ...+₂+... < NON The left side is No; the right side ≤ No.) 3.* Prove that c N. (Hint: Use Exercise 2.)
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