Inequalities¶
Recall (doc/postdoc). Not a first course.
Job: ordered-field axioms; solution sets as unions of intervals / semi-algebraic sets.
Core¶
In ordered field \(K\) (e.g. \(\mathbb{R}\)): \(<\) total, compatible with \(+\) and with \(\cdot\) by positives.
| Rule | Statement |
|---|---|
| Translation | \(a<b\Rightarrow a+c<b+c\) |
| Positive scale | \(a<b\), \(c>0\Rightarrow ac<bc\) |
| Negative scale | \(a<b\), \(c<0\Rightarrow ac>bc\) (flip) |
| Reciprocal | \(0<a<b\Rightarrow 0<1/b<1/a\) |
Squares: \(x^2\ge0\) for all \(x\in\mathbb{R}\); \(=0\) iff \(x=0\).
Interval notation: \((a,b),[a,b],(-\infty,a]\), etc. — topology of order on \(\mathbb{R}\).
Polynomial inequalities: sign chart via roots/multiplicities; later Sturm, cylindrical algebraic decomposition.
Used later for¶
- Absolute value: \(|x|<\varepsilon\) balls
- Optimization: feasible regions, KKT
- Real analysis: \(\varepsilon\)-defs, monotone convergence
- Probability: Markov/Chebyshev tail bounds
Check¶
Does \(a<b\Rightarrow a^2<b^2\) hold for all real \(a,b\)?
No — counterexample \(a=-2\), \(b=-1\): \(-2<-1\) but \(4>1\). Needs \(0\le a<b\) or compare \(|a|,|b|\).