squeeze theorem
The squeeze theorem is the trick of trapping an unruly sequence between two cooperative ones. If a sequence you cannot directly analyze is pinned above and below by two sequences that both converge to the same value, then it has no escape — it is squeezed onto that same limit.
Precisely: suppose b_n <= a_n <= c_n for all sufficiently large n, and b_n -> L and c_n -> L (the same L). Then a_n -> L as well. The middle sequence need not be monotone, need not have an obvious formula, and need not be analyzable on its own — all that matters is that it is hemmed in by the two outer sequences that share a limit.
This is one of the most practical tools in the limit toolbox, especially for sequences involving oscillation that an algebra-of-limits computation cannot directly touch. The key hypothesis is that the two bounding sequences converge to the same value; if their limits differ, no conclusion follows about the trapped sequence.
To find the limit of a_n = sin(n)/n: since -1 <= sin(n) <= 1, we have -1/n <= sin(n)/n <= 1/n. Both -1/n -> 0 and 1/n -> 0, so the squeeze theorem forces sin(n)/n -> 0, even though sin(n) itself never settles.
Squeezing converts an unmanageable oscillating sequence into a clean limit.