Let E⊆RE\subseteq\mathbb{R}E⊆R. A real number sss is the supremum of EEE if: x≤sx\le sx≤s for all x∈Ex\in Ex∈E (so sss is an upper bound ), and for every u∈Ru\in\mathbb{R}u∈R, if x≤ux\le ux≤u for all x∈Ex\in Ex∈E then s≤us\le us≤u (so sss is the least upper bound). Uniqueness of supremum: If sss and ttt are both sup⁡E\sup EsupE, then s=ts=ts=t. ...

An upper bound of a subset SSS of an ordered set (X,≤)(X,\le)(X,≤) is an element u∈Xu\in Xu∈X such that ∀s∈S, s≤u.\forall s\in S,\ s\le u.∀s∈S, s≤u.Upper bounds formalize the idea that a set lies entirely to the “left” of some point. The existence and structure of upper bounds is central to completeness and to definitions such as supremum. Examples: In (R,≤)(\mathbb{R},\le)(R,≤), the set S=(0,1)S=(0,1)S=(0,1) has upper bounds u=1u=1u=1, u=2u=2u=2, and in fact every u≥1u\ge 1u≥1. In (R,≤)(\mathbb{R},\le)(R,≤), the set S={x∈R:x<0}S=\{x\in\mathbb{R}: x<0\}S={x∈R:x<0} has upper bounds u=0u=0u=0 and every u≥0u\ge 0u≥0. In (Z,≤)(\mathbb{Z},\le)(Z,≤), the set S={n∈Z:n≤5}S=\{n\in\mathbb{Z}: n\le 5\}S={n∈Z:n≤5} has upper bound u=5u=5u=5 (and any u≥5u\ge 5u≥5).

Let f:[a,b]→Rf:[a,b]\to\mathbb{R}f:[a,b]→R be bounded and let P:a=x0<⋯<xn=bP:a=x_0<\cdots<x_n=bP:a=x0​<⋯<xn​=b be a partition. For each subinterval, define Mi:=sup⁡{f(x):x∈[xi−1,xi]}.M_i := \sup\{f(x): x\in[x_{i-1},x_i]\}.Mi​:=sup{f(x):x∈[xi−1​,xi​]}. The upper sum of fff with respect to PPP is ...

Weierstrass Approximation Theorem: Let f:[a,b]→Rf:[a,b]\to\mathbb{R}f:[a,b]→R be continuous and let ε>0\varepsilon>0ε>0. Then there exists a polynomial ppp such that sup⁡x∈[a,b]∣f(x)−p(x)∣<ε. \sup_{x\in[a,b]} |f(x)-p(x)|<\varepsilon. supx∈[a,b]​∣f(x)−p(x)∣<ε. This theorem is foundational in approximation theory: continuous functions can be uniformly approximated by simple algebraic objects (polynomials). It is also a prototype of many “density” results in functional analysis. ...

Weierstrass M-test: Let XXX be a set and let fn:X→Rf_n:X\to\mathbb{R}fn​:X→R (or C\mathbb{C}C). Suppose there exist numbers Mn≥0M_n\ge 0Mn​≥0 such that ∣fn(x)∣≤Mnfor all x∈X and all n, |f_n(x)|\le M_n \quad \text{for all } x\in X \text{ and all } n, ∣fn​(x)∣≤Mn​for all x∈X and all n, and the numerical series ∑n=1∞Mn\sum_{n=1}^\infty M_n∑n=1∞​Mn​ converges . Then the function series ∑n=1∞fn(x)\sum_{n=1}^\infty f_n(x)∑n=1∞​fn​(x) converges uniformly on XXX. In particular, it converges absolutely and uniformly: sup⁡x∈X∑n=1∞∣fn(x)∣<∞. \sup_{x\in X}\sum_{n=1}^\infty |f_n(x)| < \infty. supx∈X​∑n=1∞​∣fn​(x)∣<∞. ...

Well-ordering principle for N\mathbb{N}N: If S⊆NS\subseteq \mathbb{N}S⊆N is nonempty, then there exists m∈Sm\in Sm∈S such that m≤nm\le nm≤n for all n∈Sn\in Sn∈S. This principle is equivalent (in standard foundations) to mathematical induction and is often used to justify “choose the smallest counterexample” arguments. Proof sketch (optional): One shows that if a nonempty set S⊆NS\subseteq\mathbb{N}S⊆N had no least element, then by induction no natural number could belong to SSS, contradicting nonemptiness. ...

Let f:[a,b]→Rf:[a,b]\to\mathbb{R}f:[a,b]→R be continuous on [a,b][a,b][a,b] and differentiable on (a,b)(a,b)(a,b). Corollary: If f′(x)=0f'(x)=0f′(x)=0 for all x∈(a,b)x\in(a,b)x∈(a,b), then fff is constant on [a,b][a,b][a,b]. Connection to parent theorem: Apply the mean value theorem : for any x<yx<yx<y, there exists c∈(x,y)c\in(x,y)c∈(x,y) with f(y)−f(x)=f′(c)(y−x)=0f(y)-f(x)=f'(c)(y-x)=0f(y)−f(x)=f′(c)(y−x)=0. ...