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Problem 1. Suppose E is a given set, and On, for n ∈ N, is the...

Problem 1. Suppose E is a given set, and On, for n ∈ N, is the set defined by

On = {x ∈ Rd : d(x, E) < 1/n }.

(a) Prove that On is open.

(b) Prove that if E is compact, then m(E) = limn →∞ m(On).

(c) Would the above be true for E closed and unbounded set?

(d) Would the above be true for E open and bounded set?

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