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I'm having trouble understanding this proof from a textbook for the bias of the mean estimator using ratio estimation.

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How does $$\frac 1 {\bar{x_u}}[BV(\bar{x})-\operatorname{Cov}(\bar{x},\bar{y}) = \left(1 - \frac n N \right) \frac 1 {n\bar{x}_u}(BS_x^2-RS_xS_y) \text{ ?}$$

where:

$S_x$ and $S_y$ are the population standard deviations, $B = \frac{\bar{y}_u}{\bar{x}_u}$ and $R$ is the population correlation coefficient

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Hint: Under simple random sampling,\ $Var\{\bar{x}\}=\dfrac{1-f}{n}S_{x}^{2}$ and $Cov\{\bar{x},\bar{y}\}=\dfrac{1-f}{n}\rho S_{x}S_{y}$.

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    How does $Cov\{\bar{x},\bar{y}\}=\dfrac{1-f}{n}\rho S_{x}S_{y}$? – 2017-02-14
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    It is standard text book result in simple random sampling. See Cochran p25. – 2017-02-14