Mesh of a partition

Let PPP be a partition of [a,b][a,b][a,b] given by a=x0<⋯<xn=ba=x_0<\cdots<x_n=ba=x0<⋯<xn=b. The mesh (or norm) of PPP is ∥P∥:=max⁡1≤i≤n(xi−xi−1)=max⁡1≤i≤nΔxi.\|P\| := \max_{1\le i\le n}(x_i-x_{i-1})=\max_{1\le i\le n}\Delta x_i.∥P∥:=1≤i≤nmax(xi−xi−1)=1≤i≤nmaxΔxi.The mesh measures how fine a partition is. Many convergence statements for Riemann sums are phrased in terms of ∥P∥→0\|P\|\to 0∥P∥→0. ...

Metric space

A metric space is a pair (X,d)(X,d)(X,d) where XXX is a set and ddd is a metric on XXX, i.e. a function d:X×X→[0,∞)d:X\times X\to[0,\infty)d:X×X→[0,∞) satisfying positive definiteness, symmetry, and the triangle inequality. Metric spaces generalize Euclidean spaces and provide the setting for “analysis without coordinates.” Many results in real analysis extend to general metric spaces once stated in terms of ddd. ...

Minimum

Let (X,≤)(X,\le)(X,≤) be an ordered set and let S⊆XS\subseteq XS⊆X. An element m∈Sm\in Sm∈S is a minimum of SSS (written m=min⁡Sm=\min Sm=minS) if ∀s∈S, m≤s.\forall s\in S,\ m\le s.∀s∈S, m≤s.A minimum is a lower bound that actually lies in the set. If a minimum exists, it is unique and equals the infimum : min⁡S=inf⁡S\min S=\inf SminS=infS. ...

Mixed partial derivative

Let U⊆RkU\subseteq\mathbb{R}^kU⊆Rk be open and let f:U→Rf:U\to\mathbb{R}f:U→R be a scalar-valued function. If the partial derivative ∂f∂xj\frac{\partial f}{\partial x_j}∂xj∂f exists on a neighborhood of a point a∈Ua\in Ua∈U, and if ∂f∂xj\frac{\partial f}{\partial x_j}∂xj∂f is partially differentiable with respect to xix_ixi at aaa, then the mixed partial derivative is ...

Modulus (absolute value) on ℂ

For z=a+bi∈Cz=a+bi\in\mathbb{C}z=a+bi∈C, the modulus (or absolute value) of zzz is ∣z∣:=a2+b2=zz‾.|z|:=\sqrt{a^2+b^2}=\sqrt{z\overline{z}}.∣z∣:=a2+b2=zz.The modulus makes C\mathbb{C}C into a normed space and induces the standard metric d(z,w)=∣z−w∣d(z,w)=|z-w|d(z,w)=∣z−w∣. It is the complex analogue of absolute value and is crucial for convergence of complex sequences and series. ...

Monotone Convergence Theorem (for sequences)

Monotone Convergence Theorem (sequences): If (xn)(x_n)(xn) is a monotone increasing sequence in R\mathbb{R}R that is bounded above , then (xn)(x_n)(xn) converges and lim⁡n→∞xn=sup⁡{xn:n∈N}.\lim_{n\to\infty} x_n = \sup\{x_n:n\in\mathbb{N}\}.limn→∞xn=sup{xn:n∈N}. Similarly, a monotone decreasing sequence that is bounded below converges to its infimum . ...

Monotone functions are Riemann integrable

Monotone functions are Riemann integrable: If f:[a,b]→Rf:[a,b]\to\mathbb{R}f:[a,b]→R is bounded and monotone (nondecreasing or nonincreasing), then fff is Riemann integrable on [a,b][a,b][a,b]. This is a key example showing that Riemann integrability does not require continuity; controlled discontinuities are allowed. Proof sketch: Assume fff is nondecreasing. For a partition PPP with mesh ∥P∥\|P\|∥P∥, one can estimate U(f,P)−L(f,P)≤(f(b)−f(a)) ∥P∥. U(f,P)-L(f,P)\le (f(b)-f(a))\,\|P\|. U(f,P)−L(f,P)≤(f(b)−f(a))∥P∥. Choosing ∥P∥<ε/(f(b)−f(a))\|P\|<\varepsilon/(f(b)-f(a))∥P∥<ε/(f(b)−f(a)) (or handling the constant case separately) forces U−L<εU-L<\varepsilonU−L<ε, proving integrability. ...

Monotone sequence

A real sequence (an)n∈N(a_n)_{n\in\mathbb{N}}(an)n∈N is monotone if it is either: nondecreasing: an+1≥ana_{n+1}\ge a_nan+1≥an for all nnn, or nonincreasing: an+1≤ana_{n+1}\le a_nan+1≤an for all nnn. Monotone sequences are central in real analysis because bounded monotone sequences converge in R\mathbb{R}R (monotone convergence theorem for sequences). ...

Monotone subsequence lemma

Monotone subsequence lemma: Every sequence (xn)(x_n)(xn) in R\mathbb{R}R has a monotone subsequence ; i.e., there exists a subsequence that is either nondecreasing or nonincreasing. This lemma is a key combinatorial tool in real analysis and is often used to extract structured subsequences before applying completeness or compactness arguments. Proof sketch: Call an index nnn a “peak” if xn≥xmx_n\ge x_mxn≥xm for all m≥nm\ge nm≥n. If there are infinitely many peaks, selecting them yields a nonincreasing subsequence. If there are only finitely many peaks, then beyond some index every term is followed by a larger term; one can inductively choose indices n1<n2<⋯n_1<n_2<\cdotsn1<n2<⋯ with xn1<xn2<⋯x_{n_1}<x_{n_2}<\cdotsxn1<xn2<⋯, giving a strictly increasing subsequence. ...

Multiple (Riemann) integral over a rectangle

Let R=∏j=1n[aj,bj]⊆RnR=\prod_{j=1}^n [a_j,b_j]\subseteq \mathbb{R}^nR=∏j=1n[aj,bj]⊆Rn be a rectangle and let f:R→Rf:R\to\mathbb{R}f:R→R be bounded. A partition PPP of RRR is a finite collection of subrectangles whose union is RRR and whose interiors are pairwise disjoint (typically produced by partitioning each coordinate interval). For each subrectangle Q⊆RQ\subseteq RQ⊆R, let MQ=sup⁡x∈Qf(x),mQ=inf⁡x∈Qf(x),vol⁡(Q)=∏j=1n(side length in coordinate j).M_Q=\sup_{x\in Q} f(x), \qquad m_Q=\inf_{x\in Q} f(x), \qquad \operatorname{vol}(Q)=\prod_{j=1}^n (\text{side length in coordinate }j).MQ=supx∈Qf(x),mQ=infx∈Qf(x),vol(Q)=∏j=1n(side length in coordinate j). Define the upper sum and lower sum U(f,P)=∑Q∈PMQ vol⁡(Q),L(f,P)=∑Q∈PmQ vol⁡(Q).U(f,P)=\sum_{Q\in P} M_Q\,\operatorname{vol}(Q), \qquad L(f,P)=\sum_{Q\in P} m_Q\,\operatorname{vol}(Q).U(f,P)=∑Q∈PMQvol(Q),L(f,P)=∑Q∈PmQvol(Q). ...