Inequality of Pearson correlation coefficient Let $x_1,\ldots x_n,y_1,\ldots y_n$ be reals and $\bar{x},\bar{y}$ the aritmetic mean of numbers $x_1,\ldots x_n$ and $y_1,\ldots y_n$ respectively. How can I show that $$-1\leq \dfrac{\sum_{i=1}^n (x_i-\bar{x})(y_1-\bar{y})}{\sqrt{\sum_{i=1}^n(x_i-\bar{x})^2\sum_{i=1}^n(y_i-\bar{y})^2}}$$ Looks a bit like Cauchy (it gives me that the expression is at most one) but I was unable to find the proof.

Is just the relationship between a scalar product (numerator), the norm of the vectors (denominator) and the cosinus of the angle.
