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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 :)

Solution Summary

There are two real analysis proofs here, one regarding unions and supremum and one regarding measurable functions.

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