Definition: Sequence

A sequence is a function whose domain is \(\mathbb{N}\).

[2.2.3] Definition: Convergence of a Sequence

A sequence \((a_n)\) converges to a real number \(a\) if, for every positive number \(\epsilon\), there exists an \(N \in \mathbb{N}\) such that whenever \(n \geq N\) it follows that \(|a_n - a| \leq \epsilon\).

Intuitively, this can also be written as \(\displaystyle \lim_{n\to\infty}a_n = a\). So as \(n\) gets larger, \(a_n\) gets closer to \(a\). But we define it this way because we want to be precises. Next, we define what \(|a_n - a| \leq \epsilon\) is?

Definition: Epsilon Neighborhood of \(a\)

Given a real number \(a \in \mathbb{R}\) and a positive number \(\epsilon > 0\), the set
$$ \begin{align*} V_{\epsilon}(a) = \{x \in \mathbb{R}: |x - a| \leq \epsilon\} \end{align*} $$
is called the \(\epsilon\)-neighborhood of \(a\).

\(V_{\epsilon}(a)\) is an interval around \(a\) consisting of all the points whose distance from \(a\) is less than \(\epsilon\). So the earlier definition of convergence asserts that after some \(N\), the sequence ends up in that \(\epsilon\) neighborhood of \(L\).

Definition: Convergence of a Sequence: (Topological Version)

A sequence \((a_n)\) converges to \(a\) if, given any \(\epsilon\)-neighborhood \(V_{\epsilon}(a)\) of \(a\), there exits a point in the sequence after which all of the terms are in \(V_{\epsilon}(a)\). In other words, every \(\epsilon\)-neighborhood contains all but a finite number of the terms of \((a_n)\).

One note here as mentioned in the book. This value of \(N\) will really depend on the value of \(\epsilon\) we choose. If we choose a small \(\epsilon\), then we expect \(N\) to be larger.

Definition: Uniqueness of Limits

The limit of a sequence, when it exists must be unique.

This will certainly require a proof!

Definition: Divergence

A sequence that doesn't converge is said to diverge.

We know that a sequence converges to \(L\) if for all \(\epsilon > 0\), there exists \(N \in \mathbb{N}\) such that some tail of the sequence in the interval \((L-\epsilon, L+\epsilon)\).

The negation of this statement is as follows:

A sequence does not converge to \(L\) if there exists some \(\epsilon\) such that for all tails of the sequence, the tail will not be inside the interval \((L-\epsilon, L+\epsilon)\). So it doesn’t matter where the tail starts, for that specific \(\epsilon\), this tail will never be fully contained in that interval. In other words, for all \(N \in \mathbb{N}\), there is some \(n > N\) such that \(n \not\in (L-\epsilon, L+\epsilon)\). (So no matter what \(N\) we choose, we can always pick some \(n > N\) where the sequence will no longer be in that interval)

Now, a sequence diverges if it doesn’t converge for any \(L\).

As a quick example, take \(\{1,2,3,4,\cdots\}\). Suppose for it converges to \(100\). Pick \(\epsilon = 1\). The converges definition says that there exists an \(N\) such that for any \(n > N\), \(n\) falls into \((99,101)\). We will show that this is false by showing that for every \(N\), there is some \(n > N\), where \(n\) doesn’t fall in the interval. Pick \(n = N + 101\), then for any \(N\), we can see that \(n = N+101\) is always outside the interval. Hence, the sequence doesn’t converge to \(L=100\).

[2.3.1] Definition: Bounded Sequences

A sequence \(x_n\) is bounded if there exists a number \(M > 0\) such that every term in the sequence \(|x_n| \leq M\) for all \(n \in \mathbb{N}\).

For the proof see this.

[2.3.2] Convergent Sequences

Every convergent sequence is bounded.

For the proof see this.

Definition: Increasing, Decreasing and Monotone Sequences

A sequence \(a_n\) is increasing if \(a_n \leq a_{n+1}\) for all \(n \in \mathbb{N}\) and decreasing if \(a_n \geq a_{n+1}\) for all \(n \in \mathbb{N}\). A sequence is monotone if it is either increasing or decreasing.

[2.4.2] Monotone Convergence Theorem

If a sequence is monotone and bounded, then it converges.

For the proof see this.

[2.6.1] Definition: Cauchy Sequences

A sequence \(a_n\) is called a Cauchy sequence if, for every \(\epsilon > 0\), there exists an \(N \in \mathbb{N}\) such that whenever \(m, n \geq N\) it follows that \(|a_n - a_m| < \epsilon\).

References