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1) show that every \(\displaystyle f_{n}\) is a simple summable function.

So Firstly I need to show I can write is as a linear combination of indicator functions. Not sure how to proceed.

2)Show \(\displaystyle (f_{n}) \)is a cauchy sequence with the metric which is the integral from 0 to 1 of |f-g|.

I think the key to this question is what is the relation between the difference of the integer parts and the integer part of the difference.

3) show there is no simple summable f on [0,1] such that \(\displaystyle f_{n}\) converges to f in the above metric

Thanks