2.3 More on Infinite Limits

We can also make the idea of a sequence getting increasingly more positive (or more negative) more precise via the idea of divergence to \(\pm\infty\). We present the definition for ‘positive’ \(\infty\) here.

Definition 2.2: (Divergence to Infinity)
A real sequence \((a_n)_{n\in\mathbb{N}}\) diverges to \(\infty\) if \[\forall M \in \mathbb{R}, \; \exists N \in \mathbb{N} \; \text{such that}\; \forall n \geq N, \; a_n > M.\]

There is also a corresponding version of the ‘algebra of limits’ for divergence to \(\pm\infty\) (see Theorem 2.4). This version has been stolen and adapted from an old set of lecture notes (ones from 2016 to be precise!), so some of these results may not appear in the current lecture notes. Consequently, you won’t have seen the proofs for these results, so you can’t use them in Tutorial Question 4 on Exercise Sheet Six.3

Theorem 2.4: (Algebra of Infinite Limits)

Let \((a_n)_{n\in\mathbb{N}}\) and \((b_n)_{n\in\mathbb{N}}\) be real sequences.

  1. If \(a_n \to \infty\) and \(y_n \to \infty\) as \(n \to \infty\), then \(a_n + b_n \to \infty\) as \(n\to\infty\).
  2. If \(a_n \to \infty\) and \(c > 0\), then \(ca_n \to \infty\) as \(n \to \infty\).
  3. If \(a_n \to \infty\) and \(y_n \to \infty\) as \(n \to \infty\), then \(a_nb_n \to \infty\) as \(n\to\infty\).
  4. We have \(a_n \to \infty\) as \(n \to \infty\) if and only if \(-a_n \to -\infty\) as \(n \to \infty\).
  5. For \(a_n \neq 0\; \forall n \in \mathbb{N}\), if \(a_n \to \infty\) as \(n \to \infty\), then \(\frac{1}{a_n} \to 0\) as \(n \to \infty\).
  6. If If \(a_n \to 0\) as \(n\to\infty\) and \(a_n > 0\; \forall n \in \mathbb{N}\), then \(\frac{1}{a_n} \to \infty\) as \(n \to \infty\).

  1. This question asks you to prove parts of this theorem, so using the theorem to prove the theorem is purely circular reasoning, and should absolutely be avoided.↩︎