MATHEMATICAL ANALYSIS 01
Bounds, Intervals and Absolute Value
Analysis begins with precise language for location, distance and size.
Intervals
The open interval \((a,b)\) contains values satisfying \(a\lt x\lt b\). The closed interval \([a,b]\) also contains its endpoints.
\[x\in[a,b]\iff a\le x\le b\]
Bounds
A set \(S\) is bounded above if some number \(M\) satisfies \(x\le M\) for every \(x\in S\). It is bounded below if some \(m\) satisfies \(m\le x\).
The least upper bound is the supremum; the greatest lower bound is the infimum.
Absolute value as distance
\[|x-a|=\text{distance from }x\text{ to }a\]
Thus \(|x-a|\lt\delta\) means \(a-\delta\lt x\lt a+\delta\). This language becomes central in rigorous limits.
Triangle inequality
\[\boxed{|x+y|\le |x|+|y|}\]
It controls distances and is repeatedly used in proofs.