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I am looking for advice on if I have approached this problem correctly: Problem

Here is what I have come up with: enter image description here

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    Looks good to me. To make the last step more rigorous and formal, you may have to introduce some $\varepsilon$ and show that with an appropriate choice of $N$ the last sum is less than $\varepsilon$ for all $n,m>N$. – 2017-02-21

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assume $0

$\int_0^1 |f_n(t) - f_m(t)|\ dt = 3\frac {n-m}{2nm}< \frac {3}{2} \frac {1}{m}$