Squeeze theorem
Statement

The squeeze theorem is formally stated as follows.

Theorem — Let I be an interval containing the point a. Let g, f, and h be functions defined on I, except possibly at a itself. Suppose that for every x in I not equal to a, we have g (x) ≤ f (x) ≤ h (x) {\displaystyle g(x)\leq f(x)\leq h(x)} and also suppose that lim x → a g (x) = lim x → a h (x) = L. {\displaystyle \lim _{x\to a}g(x)=\lim _{x\to a}h(x)=L.} Then lim x → a f (x) = L. {\displaystyle \lim _{x\to a}f(x)=L.}

This theorem is also valid for sequences. Let (a), (c) be two sequences converging to ℓ, and (b) a sequence. If ∀ n ≥ N, N ∈ N {\displaystyle \forall n\geq N,N\in \mathbb {N} } we have a ≤ b ≤ c, then (b) also converges to ℓ.
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