be a decreasing sequence of nonempty, closed subsets of 3. Let E 2 E2 ... 2...

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be a decreasing sequence of nonempty, closed subsets of 3. Let E₁ 2 E₂2 ... 2 Ek 2 R". (a) Prove that if E₁ is compact, then En ‡0. n=1 (b) Find an example of a sequence as above with the property that En = 0. (Hint: In n=1 light of (a), none of the E, can be compact sets. Since each E; is closed by assumption, each of the E, must be unbounded. Also, since you are just finding one example, you can choose the E, CR.) be a decreasing sequence of nonempty, closed subsets of 3. Let E₁ 2 E₂2 ... 2 Ek 2 R". (a) Prove that if E₁ is compact, then En ‡0. n=1 (b) Find an example of a sequence as above with the property that En = 0. (Hint: In n=1 light of (a), none of the E, can be compact sets. Since each E; is closed by assumption, each of the E, must be unbounded. Also, since you are just finding one example, you can choose the E, CR.)

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3 Let Ej 26 2 262 be a decreasing sequense of honemply closed Subsets of RM a prove that if Es i... View the full answer