Why is the correlation coefficient between -1 and 1?

There are quite a few variants of the Cauchy-Schwarz inequality and I'm saying this because there is one related to statistics as well and it goes like this

Answer by Daeyoung Lim:

There are quite a few variants of the Cauchy-Schwarz inequality and I'm saying this because there is one related to statistics as well and it goes like this:
[math]\operatorname{Cov}^{2}\left(X,Y\right) \le \operatorname{Var}\left(X\right)\operatorname{Var}\left(Y\right)[/math]
It follows from this Cauchy-Schwarz inequality that the correlation coefficient is between -1 and 1.
[math]-\sqrt{\operatorname{Var}\left(X\right)\operatorname{Var}\left(Y\right)} \le \operatorname{Cov}\left(X,Y\right) \le \sqrt{\operatorname{Var}\left(X\right)\operatorname{Var}\left(Y\right)}[/math]
Therefore,
[math]-1 \le \frac{\operatorname{Cov}\left(X,Y\right)}{\sqrt{\operatorname{Var}\left(X\right)\operatorname{Var}\left(Y\right)}}\le1[/math]
The term in the middle is exactly the correlation coefficient!
Actually the covariance is the inner product between two random variables and the standard deviation is the norm of a random variable. If we denote the inner product [math]<X,Y>[/math] and the norm [math]|X|[/math], then the usual Cauchy-Schwarz inequality still holds:
[math]<X,Y>^{2}\le |X|^{2}|Y|^{2}.[/math]
But then the inner product is also defined as follows:
[math]<X,Y>=|X||Y|\cos\left(\theta\right)[/math]
The correlation coefficient is in fact the cosine of the angle between two variables:
[math]\operatorname{Corr}\left(X,Y\right) =\frac{<X,Y>}{|X||Y|}=\cos\left(\theta\right)[/math]
It's obvious that cosine ranges only from -1 to 1, right?

Why is the correlation coefficient between -1 and 1?

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