Definition: Sequence
A sequence is a function whose domain is \(\mathbf{N}\).
[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 \(N \in \mathbf{N}\) such that whenever \(n \geq N\) it follows that \(|a_n - a| \leq \epsilon\).
This can also be written as \(\displaystyle \lim_{n\to\infty}a_n = a\). What is this saying? After a certain number of terms, say after \(N\), all the points in the sequence will converge to a real number \(a\). Defining these points with \(|a_n - a| \leq \epsilon\) is explained in the next definition.
Definition: Epsilon Neighborhood of \(a\)
Given a real number \(a \in \mathbf{R}\) and a positive number \(\epsilon > 0\), the set
$$
\begin{align*}
V_{\epsilon}(a) = \{x \in \mathbf{R}: |x - a| \leq \epsilon\}
\end{align*}
$$
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 \(\epsilon\).
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.
[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 \(n \in \mathbf{N}\).
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 \(n \in \mathbf{N}\) and decreasing if \(a_n \geq a_{n+1}\) for all \(n \in \mathbf{N}\). 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 \(N \in \mathbf{N}\) such that whenever \(m, n \geq N\) it follows that \(|a_n - a_m| < \epsilon\).
References: