2.2 The Sandwich/Pinching/Squeeze Theorem²

This is a way of finding the limit of a sequence if you can find two other sequences to ‘trap’ it with. It’s quite a good method for rational functions and proving statements about \(n\)-th roots.

Theorem 2.2: (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 \(\lim_{n \to \infty}b_n = L\).

There’s also a slight modification to this theorem, called the ‘Bitten Sandwich Theorem’.

Theorem 2.3: (Bitten 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 \(\exists N \in \mathbb{N}\) such that \(a_n \leq b_n \leq c_n \; \forall n \geq N\), 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 \(\lim_{n \to \infty}b_n = L\).

This just says that as long as after some \(N \in \mathbb{N}\), \((b_n)_n\) is trapped between sequences \((a_n)_n\) and \((c_n)_n\) that share a common limit, then all three sequences will share that common limit.

It is worth noting that adaptations of theorems in this way exist all across analysis. This is because when studying convergence, we don’t really care about what is happening at the start of the sequence. We only care about the “long term” behaviour.

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2. Other names for this theorem must surely exist. If you find one out in the wild, tell me, and I’ll add it to the name of this section.↩︎