PEI Math 521K key terms

For a set S of real numbers, the smallest real number that is greater than or equal to every element of S, provided such a number exists. It need not belong to S.

For a set S of real numbers, the largest real number that is less than or equal to every element of S. Unlike a minimum, it need not be an element of the set.

The property asserting that every nonempty subset of the reals bounded above has a least upper bound in the reals. It distinguishes the reals from the rationals and underlies convergence theorems.

The statement that for any real x there is a natural number n with n greater than x. It guarantees integers and rationals are unbounded and dense in the reals.

The fact that between any two distinct real numbers there exists a rational number. Equivalently, every open interval contains a rational, so rationals approximate reals arbitrarily well.

A set whose elements can be listed in a sequence indexed by the natural numbers, or which is finite. The rationals are countable; the reals are not.

A proof technique that builds an object differing from every entry of a proposed list at the corresponding position. It shows the reals are uncountable and that certain function families fail to be countable.

The inequality stating that the absolute value of a sum is at most the sum of absolute values, and more generally that a metric distance along two legs is at least the direct distance.

The estimate that the absolute difference of two magnitudes is at most the absolute value of their difference: | |a| - |b| | is at most |a - b|. Used to bound differences of limits.

A sequence (a_n) converges to L when for every positive epsilon there is an index N such that all terms past N lie within epsilon of L. It makes convergence a precise quantitative statement.

A sequence whose terms eventually become arbitrarily close to one another: for every positive epsilon some index beyond which all pairwise distances are below epsilon. In the reals this is equivalent to convergence.

The theorem that a monotone sequence of real numbers converges exactly when it is bounded. An increasing bounded sequence converges to its supremum; a decreasing bounded one to its infimum.

The theorem that every bounded sequence of real numbers has a convergent subsequence. In R^n it says every bounded sequence has a subsequence converging to a point of the closure.

A sequence obtained from a given sequence by deleting terms while keeping the remaining ones in their original order, indexed by an increasing sequence of natural numbers.

The largest and smallest subsequential limit values of a real sequence, equivalently the limit of the suprema and infima of its tails. They always exist in the extended reals.

A sequence possessing a real limit, meaning its terms eventually stay arbitrarily close to that limit. Convergence implies boundedness, but boundedness alone does not imply convergence.

A sequence all of whose terms lie inside some fixed interval, or more generally inside some ball of finite radius in a metric space.

A convergence test for series with nonnegative terms: if a series dominates another term by term, convergence of the larger forces convergence of the smaller, and divergence of the smaller forces divergence of the larger.

Tests comparing successive-term ratios or nth roots of terms to a threshold. A limit below one gives absolute convergence, above one gives divergence, and equal to one is inconclusive.

The property of a series whose series of absolute values converges. Absolute convergence implies ordinary convergence and permits rearrangement without changing the sum.

Convergence of a series whose absolute values do not converge. Such series can be rearranged to produce different sums, as the alternating harmonic series illustrates.

The test stating that a series whose terms alternate in sign, decrease in magnitude, and tend to zero must converge.

A series whose terms are reciprocal powers n to the minus p. It converges exactly when the exponent p exceeds one, giving the standard yardstick for comparison tests.

A function f is continuous at a point c when for every positive epsilon there is a positive delta such that inputs within delta of c produce outputs within epsilon of f(c).

A property of a function on a set where one delta works for every point: for each epsilon some delta keeps all pairs of nearby inputs within epsilon in output.

The theorem that a continuous function on a compact metric space is automatically uniformly continuous. Compactness replaces the need for a point-dependent delta.

The theorem that a continuous real-valued function on a nonempty compact set attains both a maximum and a minimum value there.

The theorem that a continuous real function on an interval takes every value between the values at the endpoints, guaranteeing existence of solutions to equations.

Convergence of a sequence of functions where the rate is independent of the point: the supremum distance between each function and the limit tends to zero.

Convergence of a sequence of functions at every individual point of the domain, where the threshold index may depend on the point chosen.

A test for uniform convergence of a series of functions: if each term is bounded by a number and the numbers form a convergent series, the function series converges uniformly and absolutely.

A criterion stating that a sequence of functions converges uniformly exactly when, for every epsilon, all sufficiently late functions differ from one another by less than epsilon everywhere on the domain.

A series of coefficients times powers of a variable. There is a radius within which it converges absolutely and outside which it diverges, with uniform convergence on closed subintervals inside.

A set equipped with a distance function that is nonnegative, symmetric, zero only on equal points, and satisfies the triangle inequality.

A metric space in which every Cauchy sequence converges to a point of the space. The reals and R^n are complete; the rationals and open intervals are not.

A subset of a metric space with the property that every open cover admits a finite subcover. In R^n this is equivalent to being closed and bounded.

The theorem characterizing compact subsets of R^n as exactly those that are closed and bounded. It links the topological definition of compactness to concrete geometric conditions.

An open cover is a family of open sets whose union contains a given set. A finite subcover is a finite subfamily that still covers the set; compactness demands one for every cover.

A property of a set in which every sequence of its points has a subsequence converging to a point of the set. In metric spaces it is equivalent to compactness.

A set that cannot be split into two nonempty disjoint pieces both open relative to the set. Intervals are the connected subsets of the real line.

For a map between normed spaces, the linear map that approximates the change in output to first order near a point, with the approximation error vanishing faster than the input displacement.

The derivative of a multivariable function with respect to one variable while all others are held fixed. Existence of partials alone does not guarantee differentiability.

The matrix of all first partial derivatives of a vector-valued function. It represents the total derivative when the function is differentiable at the point.

The theorem that a continuously differentiable map with invertible derivative at a point is locally one-to-one with a differentiable inverse near that point.

The theorem giving conditions under which an equation in several variables can be solved locally for one variable as a differentiable function of the others, requiring an invertible partial derivative block.

The theorem that a function continuous on a closed interval and differentiable inside has some interior point where the derivative equals the average rate of change over the interval.

The special case of the Mean Value Theorem where the function takes equal values at both endpoints, guaranteeing an interior point with zero derivative.

The theorem approximating a function by its Taylor polynomial and expressing the error through a remainder term, often given in Lagrange or integral form.

The limit of upper and lower Darboux sums over partitions of an interval, existing when those sums agree. It equals the area under the curve for integrable functions.

A bounded function on a closed interval is Riemann integrable exactly when for every epsilon there is a partition whose upper and lower sums differ by less than epsilon.

The pair of results linking differentiation and integration: the integral of a continuous function has derivative equal to the integrand, and the integral of a derivative recovers the net change.

An integral extended over an unbounded interval or applied to an unbounded integrand, defined as a limit of ordinary integrals over truncated regions.

A property of a function whose partial derivatives all exist and are continuous. It guarantees differentiability and is the standard hypothesis in inverse and implicit function theorems.

The set of points whose distance from a center is less than a given radius. Open balls generate the metric topology and underlie definitions of openness and limits.

A set whose complement is open, equivalently one containing all its limit points. In R^n closed sets contain the limits of their convergent sequences.

A strong form of continuity where output distances are bounded by a constant times input distances. It implies uniform continuity and is used in contraction mapping arguments.

A technique that assumes the negation of the desired statement and derives an impossibility. Common in showing irrationality, uncountability, and divergence of series.

A specific case that falsifies a general statement. Constructing one is the standard way to show a claimed implication or converse fails.

The rule that the meaning of a statement depends on the order of its universal and existential quantifiers; swapping them generally changes truth value, as with continuity versus uniform continuity.