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Squeeze Theorem for Sequences

Theorem

Let (an)n=1∞,(a_n)_{n=1}^\infty\text{,} (bn)n=1∞,(b_n)_{n=1}^\infty\text{,} and (cn)n=1∞(c_n)_{n=1}^\infty be sequences of real numbers such that an≤bn≤cna_n\leq b_n\leq c_n for all n≥1.n \geq 1\text{.} Suppose also that (an)n=1∞(a_n)_{n=1}^\infty and (cn)n=1∞(c_n)_{n=1}^\infty converge to the same limit L.L\text{.} Then (bn)n=1∞(b_n)_{n=1}^\infty also converges to L.L\text{.}

Interpretation

The squeeze theorem proves convergence by trapping: a sequence caught between two sequences with the same limit has nowhere else to go. It is the standard tool when bnb_n is hard to handle directly but easy to bound by simpler sequences.

Three sequences approaching a limit L: an upper one from above, a lower one from below, and an oscillating middle one trapped between them.

In the figure, cnc_n approaches LL from above and ana_n from below, while the middle sequence bnb_n oscillates around LL. The oscillation does not matter: bnb_n always stays between ana_n and cnc_n, and the gap between them shrinks to 00.

Both hypotheses are needed. The outer limits must be equal: bn=1+(−1)nb_n=1+(-1)^n lies between an=0a_n=0 and cn=2c_n=2 but alternates between 00 and 22. Both bounds are needed too: bn=−nb_n=-n satisfies bn≤1nb_n\le\frac{1}{n} with 1n→0\frac{1}{n}\to0, yet diverges. The inequalities only have to hold from some index on, since finitely many terms do not affect a limit. Unlike the monotone convergence theorem, the theorem needs no monotonicity, and it gives both the existence and the value of the limit.

In practice the theorem is often used in the form ∣bn−L∣≤dn|b_n-L|\le d_n with dn→0d_n\to0, which is the theorem with an=L−dna_n=L-d_n and cn=L+dnc_n=L+d_n. For example, ∣(−1)nn∣≤1n\left|\frac{(-1)^n}{n}\right|\le\frac{1}{n} gives (−1)nn→0\frac{(-1)^n}{n}\to0. The same trapping argument returns later for functions in the squeeze theorem for functions.

Key Results

  • The middle sequence (bn)n=1∞(b_n)_{n=1}^\infty must lie termwise between two convergent sequences.
  • Equality of outer limits enforces convergence of the middle sequence.
  • The theorem does not require monotonicity.

Examples

  • If −1/n≤bn≤1/n,-1/n \le b_n \le 1/n\text{,} then bn→0.b_n\to 0\text{.}
  • lim⁡n→∞(n⋅sin⁡(1/n))=1.\lim\limits_{n\to\infty}\left(n \cdot\sin(1/n)\right)= 1\text{.}

Prerequisites

Sources

Aliases: sandwich theorem for sequences Keywords: sandwich theorem, squeeze theorem Symbols: infinity

Last updated: 20 October 2026 03:02