Please explain and show all work If f and g are integrable on [a,b], then so is

Question

Please explain and show all work

If f and g are integrable on [a,b], then so is their product fg. We outline a proof. Assume that f and g are nonnegative on [a, b]. Prove that for any subinterval J of [a,b], mJ(f)mJ(g) mJ(fg) MJ(fg) MJ(f)MJ(g). Suppose that 0 f(x) M and 0 g(x) M on [a,b], and that ? is a partition of [a,b]. Prove that UP(fg) - LP(fg) M(UP(f) - LP(f)) + M(UP(g) - LP(g)). Suppose f and g are integrable and nonnegative on [a, b]. Prove that fg is integrable on [a, b]. Now eliminate the nonnegativity restriction of part (c). (Hint: Problem 11 could be useful here.)

Explanation / Answer

I have written the answer on a paper and uploaded it as 5 pictures at the following links:

Part-1: http://i.imgur.com/5JZ7DYz.jpg

PArt-2: http://i.imgur.com/9oRjClJ.jpg

Part-3: http://i.imgur.com/pVVLzwV.jpg

Part-4: http://i.imgur.com/LqvYXlZ.jpg

PArt-5: http://i.imgur.com/cdsWcTX.jpg

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