The following simple equation takes in an N-length (real) vector, and spits out a (real) number between 0 and 1. (I believe this means that it is a transformation mapping $\mathfrak{R}^N \rightarrow \mathfrak{R}^1$). This equation has the property that the answer will converge to 1, as the input elements become more and more alike.
Anyway, here is the equation. $\bf x$ is the N-dimensional input vector. $y$ is the 1-dimensional scalar output. $e$ is just the exponent operator.
$ y = \frac{e^{\frac{1}{N}\displaystyle\sum_{n=1}^N \log_e(x[n])}}{\frac{1}{N}\displaystyle\sum_{n=0}^N x[n]} $
(Notice how the denominator is just the mean of $\bf x$). This function will return $y=1$ if all the elements of the vector $x$ are equal to each other.
I am trying to do two things:
1) First, I would like to 'translate' the English statement "Show that this equation converges to 1, as all the elements in the vector $\bf x$ become more and more alike", into a mathematical statement. (Italics stressing what I want translated). For example, in English we might say "As the variable $c$ approaches infinity, etc, and we use the $\displaystyle\lim_{c \to +\infty}$ to mathematically denote that.
2) After that, I would like to show (prove), that the above equation actually does converge to 1, as the elements of the above equation $x$ become more and more similar. (I know that it does indeed converge to 1, but would like to prove it).