Thoughts on the Formation of Limits

I have recently finished writing up my notes on bounded sets in Real Analysis (I will be posting my incomplete notes for Real Analysis shortly), and what is fascinating me was how the concept of these sets led to the formation of limits. Here are my thoughts:

Suppose that $\mathbb{F}$ is an ordered field and $X$ is bounded above in $\mathbb{F}$. A number $B\in\mathbb{F}$ is called a least upper bound of $X$ if (i) $B$ is an upper bound for $X$, and (ii) if $\alpha$ is any upper bound for $X$, then $B\leq\alpha.$ Similarly, if $X$is bounded below in $\mathbb{F}$, then a number $b\in\mathbb{F}$ is called a greatest lower bound of $X$ if (i) $b$ is a lower bound for $X$, and (ii) if $\alpha$ is any lower bound for $X$, then $b\geq\alpha.$ From this definition, we can see that the set $X$ is a bounded set both above and below, and is a subset of ordered field $\mathbb{F}$. We can think of lub $X$ and glb $X$ as the closest approximation of a value $x$ that cannot be defined. The reason for having the least and greatest bounds of a set is to not find the value of $x$; however it is to find a way to get close enough to $x$. And that is essentially what a limit is.

If we have a limit from $x$ to $0$ ($\lim_{x\rightarrow0}$) for some function $f(x)$, we are relatively finding the glb $f(x)$ closest to $0$. We can then have an intuition for the limit $\lim_{x\rightarrow\infty}f(x)$. This shows that we want to get $x$ as close to $\infty$ as possible. From here, these infinite limits are classified as sequences. However, I will leave the details for you to study on your own.