Correlation cannot be more extreme than +1/-1, proof using Cauchy-Schwarz inequality

Load packages library(tidyverse) The correlation coefficient cannot exceed an absolute value of 1 This is well-known. But why is that the case? How can we proof it? This post gives one explanation using the Cauchy-Schwarz inequality. Here’s one version of the definition of correlation: [ r = \frac{\sum(\Delta x \Delta y)}{\sqrt{\sum \Delta x^2} \sqrt{\sum \Delta y^2}} ] where (\Delta x) and (\Delta y) are the differences of (x_i) and (\bar{x}), that is: (\Delta x_i = x_i - \bar{x}), and similarly for (\Delta …