Consider the function $f: \mathbb{N} \to \mathbb{R}$ where

$$ f(x) = x^2 $$

This is a simple quadratic equation you’re acquainted with, but note that our domain are only natural numbers, meaning $1, 2, 3, 4, \ldots$. The graph of this function looks like this:

But this may not be the graph of the most common version of this quadratic. Indeed, it’s more commonly defined as a function from $f: \mathbb{R} \to \mathbb{R}$. While the natural numbers are said to be countable, the real numbers are uncountable.

Understanding Limits: Targets and Approaches

Introduction

When you first learned about limits in calculus, you likely encountered two different-looking definitions that your textbook claimed were “equivalent.” One used sequences and indices, while the other used those mysterious $\delta$ and $\epsilon$ symbols. Perhaps you wondered: Why do we need two definitions for the same concept? Are they really the same?

This article explores the conceptual machinery behind limits by introducing two intuitive ideas: targets (where we want our function values to land) and approaches (how we get close to a point in the domain). We’ll see that these definitions are equivalent in $\mathbb{R}$, but for a subtle and beautiful reason that connects to the Archimedean principle and the countable structure of the natural numbers.

The outcomes of a sample space (see the article on sample spaces referenced earlier) are not necessarily numerical. However, as mathematicians, we are interested in quantities such as averages, variances, and distributions, all of which require numbers.

This is where random variables come in.

A random variable is a function that maps outcomes of a sample space to real numbers. If the sample space is denoted by $\Omega$, then a random variable is a mapping

Let $\{a_n\}_{n=m}^{\infty}$ be a sequence of reals. Think of $m$ as the starting index of the sequence. Here are some sequences with different starting indices:

For $m=1$:

$$ a_1, a_2, a_3, a_4, a_5, \ldots $$

For $m=3$:

$$ a_3, a_4, a_5, a_6, a_7, a_8, \ldots $$

Now consider a natural number $N \geq m$. A tail of our sequence $\{a_n\}_{n=m}^{\infty}$ is defined as:

$$\{a_n\}_{n=N}^{\infty}$$

Visual Illustration

Let’s take $m=1$ and consider the original sequence starting from index 1:

If you interrogate the formula for the dot product, it will gift to you all the explanations you need. You’ve seen mathematicians define the same dot product in different ways.

The Geometric Definition of the Dot Product

In the two-dimensional plane, start with two vectors at the origin. Think of our vectors as line segments that we draw starting at coordinate $(0,0)$ (your origin). The dot product of those two line segments (vectors) equals:

THROW YOUR RULER, an integral will be all you need to measure distance. In fact, the integral can measure distances your ruler cannot.

You’re an engineer tasked with measuring the distance traveled by a rollercoaster on a parabolic path. Let the function $y = x^2$ on the interval $[-1, 1]$ represent that parabola. Keep in mind that the rollercoaster doesn’t exist yet, so we cannot directly go and measure it. We only know that it will follow the shape of the above quadratic.

Two friends can race at the exact same speed, yet one may be accelerating while the other is not. Why? The key lies in the velocity vector.

This idea is mathematically subtle, but it can be captured with the right intuition.

Speed is a scalar quantity. It tells you how fast something moves, but not where it is going. Velocity, on the other hand, is a vector. It has both a magnitude, which is the speed, and a direction.

You’re a girl going abroad for university. A friend tells you: a male classmate from back home will go to the same country. A coincidence, right?

Then you hear he’s in the same city. Closer.

A friend mentions his neighborhood—you tense—he’s moving to the same area.

Finally, you learn from your landlord that you’ll have a roommate. Guess who?

There’s no place for chance anymore. Let the symbol $\epsilon$ represents the distance you look around to check for the presence of your classmate. For every possible $\epsilon$ distance you look around, no matter how small, that stalker is always there. When $\epsilon$ was the size of a country, the classmate was there. $\epsilon$ became a city, a neighborhood, even as small as a house. But you could always find that stalker around. A limit point of a set behaves almost exactly like your situation with this classmate.

As a recap, the definition of a limit point of a set states that:

A point $x\in\mathbb{R}$ is a limit point of a set $A\subset\mathbb{R}$ if

$$ \forall \varepsilon>0\ \exists y\in A\setminus{x}\quad\text{such that}\quad |y-x|<\varepsilon. $$

Now, consider a sequence of real numbers:

$$ a_1, a_2, a_3, a_4, a_5, a_6, a_7, a_8, a_9, a_{10}, \dots $$

Adapting the set definition, one might naively say: for all $\varepsilon>0$, there exists an index $i$ such that $a_i$ is very close to the proposed limit point. Formally, the naive definition would be:

In everyday mathematics, zero and one are fundamentally different.

But abstract algebra asks: what if we relax our assumptions? What happens in a ring where the additive identity and the multiplicative identity are the same? Can such a ring even exist?

The answer reveals a simple but powerful fact: if $0 = 1$ in a ring, then the ring collapses to a single element.

The Setup

A ring $R$ possesses two binary operations, commonly denoted by $+$ and $\cdot$. Assuming we are working with a ring with unity (see ring with multiplicative identity), each operation has its own identity: