Introduction
Motivation
When writing my derivative proofs series, in the introduction I assert without proof the limit laws that we would need. Here, I would like to take the time to rigorously prove these limit laws using the epsilon-delta definition, in order to provide the proper foundation for my derivative proofs series.
I remember when taking first-year calculus in university that $\epsilon - \delta$ proofs were one of the most dreaded topics. However, when reading Terence Tao’s real analysis textbook, I began to appreciate their elegance. This series gives me an excuse to revisit $\epsilon - \delta$ proofs, and helped me gain a better understanding of the foundation of modern calculus.
A Little History
The concept of a limit has been around for a very long time. As far back as the Greeks give the famous Zeno’s Paradox and Achilles and the Tortoise thought experiments, wrestling with the idea of infinite sequences. Famously, the genius of antiquity Archimedes almost discovered integration when studying quadratures of the parabola. Moreover, there were a number of unrigorous proofs for the area of a circle, which used limit-like arguments.
When Calculus was first invented by Newton and Leibniz, its foundation was largely heuristic, based on this concept of infinitesimals, as can be seen in the notation $\frac{dy}{dx}$. The issue with this approach is that the foundation of Calculus was these numbers that seemingly did not exist and could not be rigorously established. Eventually, Cauchy proposed his $\epsilon - \delta$ formulation of limits. It was quickly realized that all of calculus could be rigorously rooted in these definitions. Thus, they are what are taught today.
For further reading, I recommend The Origins of Cauchy’s Rigorous Calculus and Lectures of A Robinson’s Theory of Infinitesimals and Infinitely Large Numbers.
Fixing Circularity
A theme with many of my series has been that I often find the typical proofs given in courses or textbooks are unrigorous or circular. This series is no exception.
The proofs for the continuity of $e^x$ and $\sin(x)$ often implicitly assume their continuity within them. In the case of $e^x$, proofs typically require the continuity of $\ln(x)$, which derives its continuity from $e^x$. In the case of $\sin(x)$, arguments typically require the fact that $\lvert \sin(x) \rvert \leq \lvert x \rvert$, which is often proved using the derivative of $\sin(x)$, whose proof requires the continuity of $\sin(x)$ at $x = 0$. In this series, fix these circularities and provide a completely rigorous formulation.
Note, using the power series definition for $e^x$ and $\sin(x)$ removes all circularity, but I want to avoid this machinery.
Miscellaneous Facts
These facts will be used throughout the series.
The Triangle Inequality
The triangle inequality
\[\lvert a + b \rvert \leq \lvert a \rvert + \lvert b \rvert\]and the subtraction triangle inequality
\[\Big \lvert \lvert a \rvert - \lvert b \rvert \Big \rvert \leq \lvert a - b \rvert\]There are numerous proofs of these facts online.
Bounding Convergent Functions
If $\displaystyle \lim_{x \rightarrow a} f(x)$ exists, then $f(x)$ is bounded in a small neighborhood around $a$. Very often we will fix a particular $\epsilon > 0$ and say
\[\begin{align} &\exists \delta > 0 \quad \text{s.t.} \quad 0 < \lvert x - a \rvert < \delta \quad\implies\quad \lvert f(x) - L \rvert < \epsilon \\[10pt] &\exists \delta > 0 \quad \text{s.t.} \quad 0 < \lvert x - a \rvert < \delta \quad\implies\quad \Big \lvert \lvert f(x) \rvert - \lvert L \rvert \Big \rvert < \epsilon \\[10pt] &\exists \delta > 0 \quad \text{s.t.} \quad 0 < \lvert x - a \rvert < \delta \quad\implies\quad -\epsilon < \lvert f(x) \rvert - \lvert L \rvert < \epsilon \\[10pt] &\exists \delta > 0 \quad \text{s.t.} \quad 0 < \lvert x - a \rvert < \delta \quad\implies\quad \lvert L \rvert - \epsilon < \lvert f(x) \rvert < \lvert L \rvert + \epsilon \end{align}\]Now, we can set $\epsilon$ can be anything we want, for example $\epsilon = \frac{\lvert L \rvert}{2}$. Preprocessing this logic will save some lines in future proofs.