If capital letters are supposed to be sets, why is $N$ used as a number? For example, in this definition of a Cauchy Sequence it says:
A sequence ${p_n}$ in a metric space $X$ is said to be a Cauchy sequence if for every $\epsilon\gt 0$ there is an integer $N$ such that $d(p_n,p_m)\lt \epsilon$ if $n\ge N$ and $m\ge N$.
Is this saying that for an arbitrarily large $n$ and $m$ this is true? I'm confused on why it uses $N$ when I've been told that capital letters are supposed to be sets.
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$\begingroup$Sets are usually written as capital letters, but not necessarily. This also doesn't forbid other things being written as capital letters.
The definition of Cauchy sequence there means that one can specify an arbitrary distance and there will be a point onwards in the sequence where terms will be within that distance.
$\endgroup$ $\begingroup$While authors often use different choices of letters and capitalization to emphasize the difference in the "types" of objects that these letters refer to it is by no means obligatory.
For example we often use letters early in the alphabet ($a$, $b$, $c$) to represent constants and letters later in the alphabet ($x$, $y$, $z$) to represent variables. But if your teacher asked you to graph $f(a) = a^2 + a$ you probably wouldn't give it a second thought even though $a$ is being used as a variable and not a constant.
So basically, $N$ is a number here, not a set. And in the context of limits it's very common to use $N$ for this purpose so expect to see it again.
$\endgroup$ $\begingroup$A convention that capital letters represent sets and lower-case letters represent numbers is something that may be adopted in a book or a paper, and may even be standard in publications in a particular field of research, but it is not a universal convention in mathematics. In some contexts, capital letters near the end of the alphabet are random variables; thus $X$ may be a random variable, and one is careful about the distinction between capital $X$ and lower-case $x$ in expressions like "$F(x)=\Pr(X\le x)$" Also, in probability theory, one often uses capital $F$ for the cumulative probability distribution function given by $F(x)=\Pr(X\le x)$, and lower-case $f$ for the probability density function, for which $\Pr(X\in A) = \int_A f(x)\,dx$.
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