The Squeeze Theorem

Real Analysis · Axiom Academy

Trap a sequence between two others that close on the same limit — with — and the one in the middle has nowhere to go but L . 1. Caged Between Two Sequences Suppose three sequences satisfy for every large n . The middle one, b_n , is trapped : it can never rise above the upper wall c_n nor sink below the lower wall a_n . Now drive both walls to the same height L . The gap c_n - a_n collapses to zero, and the cell holding b_n shrinks onto the single point L — so b_n is forced there too. the trap: lower wall trapped upper wall both walls run to the same limit The squeeze is more than a picture — it follows straight from the -definition of a limit. Pick any tolerance and draw the band around L . Because and , eventually both walls enter that band and stay. From that point on the entire interval [a_n, c_n] lies inside the band — and b_n lives in that interval, so b_n is inside the band too. so for all large n — which is exactly . Here is the payoff. The sequence never converges by inspection — jumps around the interval [-1, 1] forever. But is bounded , and that is all we need to build the trap. Since , dividing by n > 0 keeps the order: Both walls are easy limits, and they agree: The trap closes on 0 , so the Squeeze Theorem hands us the answer below. You can now trap a stubborn sequence between two converging walls and read off its limit — even when a direct attack fails. Scroll up to revisit any step.

This is the written version of the interactive lesson above. See the full Real Analysis course.