1.1 Sequences and Convergence

1.1.1 Definition of Convergence

To begin, we recall the definition of a sequence:

Definition 1.1: (Sequence)

A sequence of real numbers is a function \[\begin{align*} a:\; &\mathbb{N} \longrightarrow \mathbb{R},\\ &n \longmapsto a_n. \end{align*}\]

As you saw last semester, this notation can get pretty annoying, so instead we write a sequence as \((a_n)_{n\in\mathbb{N}}\). If it’s clear from the context what set we’re indexing over, we can even just write \((a_n)_n\).

Since the natural numbers forms a countably infinite set, a sequence gives us a countably infinite list of real numbers. Sometimes it’s interesting to look at the ‘long-term’ behaviour of this list which leads on to the idea of convergence:

Definition 1.2: (Sequence Convergence)

A sequence \((a_n)_{n\in\mathbb{N}}\) converges to a real number \(L\) as \(n \longrightarrow \infty\), written as either \(a_n \longrightarrow L\), or \(\lim_{n \to \infty}a_n = L\) if \[\forall \epsilon > 0, \; \exists N = N(\epsilon) \in \mathbb{N}, \; \text{such that} \; \forall n \geq N, \; \lvert a_n - L \rvert < \epsilon.\]

Loosely speaking, this says that no matter how close you want the sequence to get to \(L\), you will always be able to find some point in the sequence after which all points in the sequence will be as close to \(L\) as desired.

1.1.2 Tests for Convergence

Using the definition to prove sequence convergence (or otherwise) for every possible sequence is incredibly tedious. So what we really want is some criteria or tests which make these proofs much easier. You’ll find some of these tests below.

The first test is the sandwich (or pinching or squeeze) theorem:

Theorem 1.1: (Sandwich Theorem)

Suppose that \((a_n)_{n\in\mathbb{N}}\, , \, (b_n)_{n\in\mathbb{N}}\, , \, (c_n)_{n\in\mathbb{N}}\) are real sequences. If \(a_n \leq b_n \leq c_n \quad \forall n \in \mathbb{N}\), and \(\exists L \in \mathbb{R}\) such that \[\begin{align*} \lim_{n \to \infty} a_n = L = \lim_{n \to \infty} c_n, \end{align*}\] then \(\lim_{n \to \infty}b_n = L\).

In words, this says that if you can ‘trap’ a sequence between two other sequences converging to a common limit, then all three sequences involved will converge to the same limit.

The second test is known as the Growth Factor Test, and aims to determine convergence by comparing the ratio of successive terms in a sequence:

Theorem 1.2: (Growth Factor Test)

Let \((a_n)_{n\in\mathbb{N}}\) be a real sequence with \(a_n>0 \; \forall n\in\mathbb{N}\), and with \[\lim_{n\to\infty} \frac{a_{n+1}}{a_n} = r.\] Then:

• If \(r < 1\), \(a_n \to 0\) as \(n \to \infty\).
• If \(r > 1\), \(a_n \to \infty\) as \(n \to \infty\).
• If \(r = 1\), the test is inconclusive.

Note that this test won’t work if the terms of the sequence \((a_n)_n\) are given by rational functions (i.e. ratios of polynomials). Can you see/prove why?

1.1.3 Subsequences

The two tests above are great at proving convergence, but is there a quick way of proving non-convergence? As you’ve seen before, we can do this using subsequences:

Definition 1.3: (Subsequence)

Let \((a_n)_{n \in \mathbb{N}}\) be a real sequence, and let \((n_k)_{k\in\mathbb{N}}\) be a strictly increasing sequence. Then \((a_{n_k})_{k\in\mathbb{N}}\) is called a subsequence of \((a_n)_{n\in\mathbb{N}}\).

Using these subsequences to prove non-convergence relies on the contrapositive of the following proposition:

Proposition 1.3:

If a real sequence \((a_n)_n\) converges to a limit \(L\), then all subsequences \((a_{n_k})_k\) of \((a_n)_n\) also converge to \(L\).

Namely, if there exists two subsequences converging to different limits, then the original sequence does not converge!

1.1.4 Limits Superior and Inferior

It is not always the case that the limit of a sequence exists — take for example the sequence \(\left((-1)^n\right)_{n\in\mathbb{N}}.\) But there are two ‘limiting’ objects which we can still talk about. These are the limit superior and limit inferior of a sequence, and can be thought of as ‘eventual’ bounds on a sequence:

Definition 1.4: (Limits Superior and Inferior)

For a sequence \((a_n)_{n\in\mathbb{N}}\), we define the limits superior and inferior to be \[\limsup_{n \to \infty} a_n := \lim_{k\to\infty}\sup_{n\geq k}a_n \;\, \text{and} \;\, \liminf_{n \to \infty} a_n := \lim_{k\to\infty}\inf_{n\geq k}a_n.\] If the sequence \((a_n)_{n\in\mathbb{N}}\) is unbounded above, we say that \(\limsup_{n \to \infty} a_n = +\infty\). If \((a_n)_{n\in\mathbb{N}}\) is unbounded below, we set \(\liminf_{n \to \infty} a_n = -\infty.\)

Alternatively, we can think of \(\limsup_{n\to\infty} a_n\) and \(\liminf_{n \to \infty}a_n\) as being the largest and smallest possible limits of any subsequence of \((a_n)_{n\in\mathbb{N}}\) respectively.