We all know from finance and statistics lectures that the correlation coefficient (Rho) between (realizations) of two (random) variables must always lie in a range between -1 and +1. It is also quite trivial to prove that Rho becomes +1 in case of perfect positive correlation and -1 in case of perfect negative correlation. (Simply suppose a linear relationship Y=aX+b or Y=ax-b and substitute the respective values in the formula of the correlation coefficent, which is: Cov(X,Y)/s(X)s(Y)).
But when it comes to prove that Rho must always lie between (and not on) -1 and +1, there is no linear relationship that enables a comfortable proof. Interestingly, most web pages covering the correlation coefficient mention this issue, but only refer to the fact that the proof was an application of the Cauchy-Schwarz Inequality and leave the proof to the reader. I will not work through the proof here either, but at least I provide you with a useful link: The full proof is published on the webpage of Rice University's Department of Statistics: Here it is.
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