1. ## A limit

For any integrable function on $\displaystyle [0,1]$ compute the value of $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,\frac{1}{n}\sum\limits_{k=0}^{n-1}{(n-k)\int_{\frac{k}{n}}^{\frac{k+1}{n}}{f(x)\,dx}}.$

2. Write $\displaystyle \sum\limits_{k=0}^{n-1}{(n-k)\int_{\frac{k}{n}}^{\frac{k+1}{n}}{f}}=\sum\limi ts_{k=0}^{n-1}{\sum\limits_{j=k+1}^{n}{\int_{\frac{k}{n}}^{\fr ac{k+1}{n}}{f}}}$ and reverse summation order.

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After reversing summation order we have,

$\displaystyle \sum\limits_{k=0}^{n-1}{\sum\limits_{j=k+1}^{n}{\int_{\frac{k}{n}}^{\fr ac{k+1}{n}}{f}}}=\sum\limits_{j=1}^{n}{\sum\limits _{k=0}^{j-1}{\int_{\frac{k}{n}}^{\frac{k+1}{n}}{f}}}=\sum\li mits_{j=1}^{n}{\int_{0}^{\frac{j}{n}}{f}}.$

Thus, the limit becomes $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,\frac{1}{n}\sum\limits_{j=1}^{n}{\int_{0}^{\fr ac{j}{n}}{f}}=\int_0^1\int_0^yf(x)\,dx\,dy,$ and the rest follows by reversing the integration order. $\displaystyle \blacksquare$