1.1 Sequences

1.1.1 Two Useful Theorems

Last week, we were introduced to the idea of sequences and what it means for a sequence to converge. The definition of this is repeated below.

Definition 1.1: (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 \[\forall \epsilon > 0, \; \exists N = N(\epsilon) \in \mathbb{N}, \; \text{such that} \; \forall n \geq N, \; \lvert a_n - L \rvert < \epsilon.\]

Using this definition, we can establish two theorems which will really help us when looking for further results about sequences.

Theorem 1.1: (Preservation of Non-Strict Inequalities)

Let \((a_n)_{n\in\mathbb{N}}\) and \((b_n)_{n\in\mathbb{N}}\) be sequences and let \(L,M \in \mathbb{R}\) be such that \(a_n \to L\) and \(b_n \to L\) as \(n \to \infty\). If \(a_n \leq b_n \; \forall n \in \mathbb{N}\), then \(L \leq M\).

There are two good uses for this theorem. The first says that non-negative sequences should have non-negative limits (which is something you might expect). Before we state the second, we mention one more thing, which is not true:
Non-theorem:

Let \((a_n)_{n\in\mathbb{N}}\) and \((b_n)_{n\in\mathbb{N}}\) be sequences and let \(L,M \in \mathbb{R}\) be such that \(a_n \to L\) and \(b_n \to L\) as \(n \to \infty\). If \(a_n < b_n \; \forall n \in \mathbb{N}\), then \(L < M\).

To see why this is false, consider the sequences defined by \(a_n = 1 - \frac{1}{n}\) and \(b_n = 1\). We note that each \(a_n\) is strictly less than each corresponding \(b_n\), but we find that \[\lim_{n \to \infty} a_n = 1 = \lim_{n \to \infty} b_n.\]

The second reason why Theorem 1.1 is so important, is that it gives us this second theorem1:
Theorem 1.2: (Uniqueness of Limits)

If \((a_n)_{n\in\mathbb{N}}\) is convergent with \(a_n \to L\) and \(a_n \to M\) as \(n \to \infty\), then \(L = M\).

1.1.2 Bounded Sequences

Much like we have done with sets, we can formulate a definition which allows us to ‘trap’ sequences.

Definition 1.2: (Bounded Sequence)
A sequence \((a_n)\) is bounded if there exists \(M \in \mathbb{R}\) such that \(\lvert a_n \rvert \leq M\).

If you prefer to think diagramatically, this says we can trap the sequence within a strip of width \(2M\) centred round \(0\). More importantly, this leads to the idea that all convergent sequences are bounded. Note that this is equivalent to saying that if a sequence is not bounded, then it is not convergent.

1.1.3 Algebra of Limits

Using the definition to prove all limits would be an incredibly boring way to go through this course. Luckily, there are a few general results we can prove which make our lives so much easier. This is known as the algebra of limits (AoL).

Theorem 1.3: (Algebra of Limits)

Let \(A,B,c \in \mathbb{R}\) and let \((a_n)\) and \((b_n)\) be sequences with \(a_n \to A\) and \(b_n \to B\) as \(n \to \infty\). Then:

  1. \(\lim_{n \to \infty} (a_n + b_n) = A + B\),
  2. \(\lim_{n \to \infty} (ca_n) = cA\),
  3. \(\lim_{n \to \infty} (a_n b_n) = AB\),
  4. If \(b_n \neq 0 \; \forall n \in \mathbb{N}\) and \(B \neq 0\), \(\lim_{n \to \infty} \frac{a_n}{b_n} = \frac{A}{B}\).

  1. Feel free to ignore this footnote, but there are areas of maths where limits are not unique. This is usually in the realm of topology, which you can take in Year 3 (MA30055). Luckily for us, everything behaves nicely, and our limits are unique.↩︎