Let E,F⊆RE,F\subseteq\mathbb{R}E,F⊆R be nonempty and bounded above/below where needed, and let c∈Rc\in\mathbb{R}c∈R. Order properties: If E⊆FE\subseteq FE⊆F and both are bounded above, then HAHAHUGOSHORTCODE515s1HBHBE≤sup⁡Fsup E\le \sup FHAHAHUGOSHORTCODE515s1HBHBE≤supF. If E⊆FE\subseteq FE⊆F and both are bounded below, then HAHAHUGOSHORTCODE515s2HBHBE≥inf⁡Finf E\ge \inf FHAHAHUGOSHORTCODE515s2HBHBE≥infF. Translation: ...

Alternating Series Test: Let (bn)(b_n)(bn​) be a decreasing sequence of nonnegative real numbers with bn→0b_n\to 0bn​→0. Then the alternating series ∑n=1∞(−1)n−1bn\sum_{n=1}^\infty (-1)^{n-1} b_n∑n=1∞​(−1)n−1bn​ converges . ...

Archimedean property of R\mathbb{R}R: For every x∈Rx\in\mathbb{R}x∈R there exists n∈Nn\in\mathbb{N}n∈N such that n>xn>xn>x. Equivalently, for every ε>0\varepsilon>0ε>0 there exists n∈Nn\in\mathbb{N}n∈N such that 1/n<ε1/n<\varepsilon1/n<ε. This property links the discrete structure of N\mathbb{N}N to the continuum R\mathbb{R}R and is used constantly in ε\varepsilonε–NNN arguments, especially to choose large integers making quantities small. Proof sketch (optional): If the set N\mathbb{N}N were bounded above , it would have a supremum sss. Then s−1s-1s−1 would not be an upper bound, so some integer nnn satisfies n>s−1n>s-1n>s−1, implying n+1>sn+1>sn+1>s, contradicting that sss is an upper bound. ...

Let (K,d)(K,d)(K,d) be a compact metric space and consider C(K,R)C(K,\mathbb{R})C(K,R) with the sup metric d∞(f,g)=sup⁡x∈K∣f(x)−g(x)∣. d_\infty(f,g)=\sup_{x\in K}|f(x)-g(x)|. d∞​(f,g)=supx∈K​∣f(x)−g(x)∣. A subset F⊆C(K,R)\mathcal{F}\subseteq C(K,\mathbb{R})F⊆C(K,R) is relatively compact if its closure in (C(K,R),d∞)(C(K,\mathbb{R}),d_\infty)(C(K,R),d∞​) is compact. ...

Baire Category Theorem: If (X,d)(X,d)(X,d) is a complete metric space and U1,U2,…U_1,U_2,\dotsU1​,U2​,… are open dense subsets of XXX, then ⋂n=1∞Un\bigcap_{n=1}^\infty U_n⋂n=1∞​Un​ is dense in XXX. Equivalently, XXX is not a countable union of nowhere dense sets . ...

A topological space (in particular, a metric space ) XXX is a Baire space if for every sequence of open dense sets U1,U2,⋯⊆XU_1,U_2,\dots\subseteq XU1​,U2​,⋯⊆X, the intersection ⋂n=1∞Un \bigcap_{n=1}^\infty U_n ⋂n=1∞​Un​ is dense in XXX. ...

Banach Fixed Point Theorem (contraction mapping principle): Let (X,d)(X,d)(X,d) be a complete metric space and let T:X→XT:X\to XT:X→X be a contraction with contraction constant c∈[0,1)c\in[0,1)c∈[0,1). Then: There exists a unique fixed point x∗∈Xx^\ast\in Xx∗∈X such that T(x∗)=x∗T(x^\ast)=x^\astT(x∗)=x∗. For any starting point x0∈Xx_0\in Xx0​∈X, the iterates defined by xn+1=T(xn)(n≥0) x_{n+1}=T(x_n)\quad(n\ge 0) xn+1​=T(xn​)(n≥0) converge to x∗x^\astx∗. Quantitative error bounds hold: for all n≥0n\ge 0n≥0, d(xn,x∗)≤cn1−c d(x1,x0),d(xn,x∗)≤cn d(x0,x∗). d(x_{n},x^\ast)\le \frac{c^{n}}{1-c}\,d(x_1,x_0), \qquad d(x_n,x^\ast)\le c^n\,d(x_0,x^\ast). d(xn​,x∗)≤1−ccn​d(x1​,x0​),d(xn​,x∗)≤cnd(x0​,x∗). This theorem is one of the main uses of completeness: it turns a global “shrinking” hypothesis into existence and uniqueness of solutions of T(x)=xT(x)=xT(x)=x. ...

Basic properties of lim sup⁡\limsuplimsup and lim inf⁡\liminfliminf: Let (an)(a_n)(an​) be a real sequence and define sn=sup⁡k≥nak,in=inf⁡k≥nak. s_n=\sup_{k\ge n} a_k,\qquad i_n=\inf_{k\ge n} a_k. sn​=supk≥n​ak​,in​=infk≥n​ak​. Then: ...

A function f:X→Yf:X\to Yf:X→Y is bijective if it is injective and surjective ; equivalently, ∀y∈Y, ∃! x∈X such that f(x)=y,\forall y\in Y,\ \exists!\ x\in X\ \text{such that}\ f(x)=y,∀y∈Y, ∃! x∈X such that f(x)=y, where ∃!\exists!∃! means “there exists exactly one.” Bijectivity is the precise condition for the existence of an inverse function f−1:Y→Xf^{-1}:Y\to Xf−1:Y→X satisfying f−1(f(x))=xf^{-1}(f(x))=xf−1(f(x))=x and f(f−1(y))=yf(f^{-1}(y))=yf(f−1(y))=y. ...

Bolzano–Weierstrass Theorem: If (xn)(x_n)(xn​) is a bounded sequence in Rk\mathbb{R}^kRk, then there exists a subsequence (xnj)(x_{n_j})(xnj​​) and a point x∈Rkx\in\mathbb{R}^kx∈Rk such that xnj→xas j→∞.x_{n_j}\to x \quad \text{as } j\to\infty.xnj​​→xas j→∞. ...