Bounding & Squeezing

Trap the answer between two walls.

To show an integer quantity equals $n$ : prove it is $\ge n$ and $\le n$ . The two halves are often entirely different arguments, and realizing this splits a hard problem into two easier ones.

In inequalities, every bound must come with its equality case: a bound that nothing achieves answers no question. State when equality holds, always.

Where it appears on Lemma: Level 4 AM–GM, Level 7 Cauchy–Schwarz.

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More techniques

Similarity & RatiosAngles give you lengths.
Classical InequalitiesEquality tells you where to aim.
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