Real analysis proofs
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1). If g_n = Sup f_n, then prove that ( g_n)^-1 ( ( alpha, infinity] ) = union ( n = 1 to infinity) (f_n)^-1((alpha,infinity]).
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2). Prove that y(x) = inf f_n(x) is a measurable function if all f_n(x) are measurable.
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Please I want very detailed proofs, justify every step and prove every claim you make. Thanks :)
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Solution Summary
There are two real analysis proofs here, one regarding unions and supremum and one regarding measurable functions.
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1. Proof:
First, I claim that . For any , we know that . Since , then we can find some , such that . If not, then for all . This implies that . This is ...
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