Analytic Primer

I want to prime your intuition before presenting the formal definitions of limit, convergence, sequence, etc., which usually glazes the eyes of the more ardent student. I find this topic very interesting, in fact, I feel the Weierstrass-Bolzano Theorem to be the most interesting idea in mathematics and the primary motivation for my studies.

I hope sincerely that this essay primes both your understanding and your interest in analysis and calculus.

Sequence

We create a sequence by collecting terms, and we typically refer to the terms by an index set $n \in J$. We can assume that $J$ consists of all integers greater than one.

Sequence of Natural Numbers

The sequence $(x_n) = (1, 2, 3, 4, ...)$ lists all the natural numbers. Using the index notation, we see $x_1 = 1$, and $x_2 = 2$, and so on.

Sequence of Rational Numbers

The sequence $(r_n) = (\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, ... )$ lists all rational numbers less than one with numerator one. Using the index notation, we see $r_1 = \frac{1}{2}$, $r_2 = \frac{1}{3}$, and so on. Note that this does not generate all rational numbers; only a particular subset has been specified.

We're mostly interested in infinite sequences, i.e., sequences that do not terminate with a last or ultimate term. This occurs with our assumption that the index set $J$ consists of all integers greater than one. But we have the option of working with finite sequences; simply let $I \subset J$ consist of a finite number of elements. We can then only index a finitely long sequence.

Sequence of Natural Numbers Less than Eleven

The sequence $(a_n) = (1, 2, 3, 4, 5, 6, 7, 8, 9, 10)$ list every natural number with the index $n \le 10$. Using the index notation, we see $a_1 = 1$, $a_2 = 2$, and so on. But we also see that $a_11$ does not make sense; that index isn't provided.

Think for a few minutes why the rule I specify uses the index and not the set of natural numbers to bound the sequence.

A good question to ask is whether the terms in the sequence terminates at some value. Given a finite sequence, we can easily see that the sequence terminates at the final index. We might presume that an infinite sequence does not terminate. After all, the index set for an infinite sequence does not terminate, so we'd expect the sequence terms to continue forever.

We could, and ought to, make an important distinction between the infinitude of the sequence term indices and the limitations to the sequence term values. A sequence may have its terms' values slowly approach a specific, unique value without ever going beyond that term.

A Constant Sequence

The sequence $(c_n) = (1, 1, 1, 1, ...)$ consists of the number one repeated $n \subset J$ times, where $J$ may be an infinite index set. The limit of $(c_n)$ is one, since there's no index $n$ for which a term $c_n$ is greater than one. In general, we can show that, given the sequence $(c_n) = (c, c, c, c, ...)$ where $c$ is any number whatsoever, $(c_n)$ must have its limit as $c$. Think about why this might be true, as the concept will elucidate the theorem we'll be proving.

Convergence

In our first example above with $(x_n)$ we see that no matter what limit we pick, the values of $x_n$ always seem to "escape" the boundaries we set. By contrast, with the examples given in $(r_n)$, $(a_n)$, $(c_n)$, it seems we're able to contain the term values, never escaping beyond a certain value.

This observation allows us to propose the following informal definition.

Convergence

If the terms of a sequence become arbitrarily closer in value, then we way the sequence converges. Otherwise, we say the sequence diverges.

Sequence of Natural Numbers Diverges

The sequence $(x_n) = (1, 2, 3, 4, ...)$ diverges. To see this, let's set a parameter $\epsilon = 2$. If we look at $x_1$ and $x_2$, we see that the "space" between their values is one, i.e. less than two. Write this fact as

$$\vert x_1 - x_2 \vert < \epsilon = 2$$

However, we'd like to make the "space" between any terms arbitrarily small. In other words, it'd be great if epsilon were very close to zero, rather than a very conspicuously large two. Unfortunately, not only can we not make epsilon smaller, we need to make it arbitrarily large as the "space" between any terms can grow well beyond $\epsilon$.

The formal language involved in these demonstrations is called an "epsilon-delta" style proof; it's a cumbersome and confusing notation. I hope my prose explanations here make the key ideas clearer. Let me know!

A Sequence of Rational Numbers Converges

The sequence $(r_n) = (\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, ... )$ converges. To see this, set a parameter $\epsilon = 1$. If we look at $r_1$ and $r_2$, we see that the "space" between their values is less than one. Write this fact as

$$\vert r_1 - r_2 \vert < \epsilon = 1$$

In fact, the absolute value of the difference is $\frac{1}{6}$. We want epsilon to be arbitrarily small, so let's update the parameter to $\epsilon = \frac{1}{6}$ and look at the next pair of terms

$$\vert r_3 - r_4 \vert < \epsilon = \frac{1}{6}$$

In fact, the absolute value of the difference is now $\frac{1}{20}$. We can push $\epsilon$ ever smaller and the indices ever larger; we'll discover that the "space" never exceeds $\epsilon$, no matter how small we specify.

Limit

Now, I've written the definition in such a way as to omit any direct reference to a limit point. I did, however, use the concept of the limit to motivate the definition. That's because the ideas are intimately tied together.

Informally, we can say

Sequence Limit

If a sequence converges, then it has a limit.

By formalizing this theorem, we'll establish the analytic foundation for calculus.