role of inequalities

Beginning algebra is mostly about equalities — solving for the one value that makes two sides equal. Analysis flips this: it is mostly about inequalities, because to control a wobbly, approximate quantity you rarely know it exactly, but you can pin it between bounds. An analyst's instinct is not ‘what does this equal?’ but ‘how big can this be?’.

Inequalities are the working language of analysis: estimates and bounds, rather than exact equalities, are what nearly every argument manipulates. To show a quantity is small, you bound it above by something you can make small; to show a limit holds, you bound a distance below the chosen epsilon. The triangle inequality |a + b| <= |a| + |b| is the single most-used tool, letting you split and bound complicated expressions piece by piece.

This is why phrases like ‘it suffices to show ... < epsilon’ pervade proofs. You are not chasing the exact value of an error term — only a usable upper bound on it. The honest caveat: a bound is only as good as it is tight enough for the purpose. A crude estimate that still beats epsilon is a complete success; the goal is sufficiency, not optimality.