Graduate Calculus

Define a sequence as a function from natural numbers into a space, using indices n and A_n. Analyze boundedness, convergence, monotone behavior, and real-valued limits within metric spaces.

Define a subsequence as a∘j with an increasing j. The lecture shows that if a_n converges, every subsequence converges, and that c_n is not a subsequence of a_n.

Explore limit supremum and limit infimum for real sequences, defined via supremums of tails, with guaranteed existence, including finite values or infinity, and intuitive subsequential limits.

Learn how compactness forces a sequence to converge to x when every convergent subsequence does. The lecture covers convergence definitions, proof by contradiction, and the necessity of compactness with examples.

Define a series as the limit of partial sums of a real or complex sequence, and clarify that convergence depends on the sequence and not the starting index.

Apply the Cauchy condition to series by examining partial sums and their differences, noting that a nonzero limit implies divergence and convergence requires Cauchy behavior.

Study the convergence of positively valued series. Use the integral test with monotone partial sums and limsup ratio tests for exercises including tangent of one over n to the p.

Compare the root test and ratio test to determine convergence of series, including power series, using absolute values of terms to define the radius r.

Show that if sum a_n converges with non-negative terms, then sum sqrt(a_n)/n converges by Cauchy-Schwarz, using partial sums and a finite Hölder bound.

Define accumulation points and isolated points; establish limits via sequences and epsilon-delta, and show continuity depends on whether x0 is isolated or a limit point, and on non-interval domains.

Define the limit supremum of a function at an accumulation point via phi(delta), the delta-neighborhood supremum, and show its limit as delta approaches zero exists (possibly plus or minus infinity).

Explore lower semicontinuity and upper semicontinuity in variational problems, showing how infimum and minimizing sequences lead to a minimizer, for example via the Dirichlet energy and the Laplace equation.

Learn how open balls define metric space topology, distinguish open and closed sets via accumulation points, and prove equivalence of topological and pointwise continuity for maps between metric spaces.

Define compact sets by the finite subcover property for any open cover. Show that in R1, compact sets are closed and bounded, and every sequence has a convergent subsequence.

Explore the finite intersection property of compact sets, proving that any intersection of a family with nonempty finite intersections remains nonempty, and examine open coverings and Cantor set implications.

Show that continuous functions on compact sets attain a maximum and a minimum, using image compactness and sequential compactness. Explain how upper and lower semicontinuity guarantee extrema and practical consequences.

Show how compact domains elevate continuity to uniform continuity for any continuous function, using finite coverings, half-radius deltas, and the triangle inequality to bound f(x) and f(y) by two epsilon.

This lecture proves the diameter of compact sets via a direct two-step method: define delta(x)=max_y d(x,y), show delta is 1-Lipschitz and continuous, then maximize delta.

Explore how contraction type maps admit fixed points on compact spaces by analyzing the continuous distance function phi, its minimum, and the uniqueness argument.

See how local properties and a maximal set argument prove global constancy of a function on a connected interval, once a global minimum and local maximum are established.

In graduate calculus, explore how injective continuous self maps of the circle preserve connectedness, using a lemma about arcs to show the image of S1 equals S1, with a contradiction.

Examine how continuity strengthens from basic pointwise continuity to uniform, absolute, and Lipschitz forms using the modulus of continuity. Learn how omega(d(x,y)) bounds |f(x)-f(y)| and yields global, epsilon-like control.

Explore Lipschitz and Hölder continuity and their links to the mean value theorem, including modulus, local Lipschitz behavior, and the constant outcome for alpha greater than one on intervals.

Uniform continuity uses a delta independent of the point and defines a modulus; compact domains ensure it and preserve Cauchy sequences, enabling extensions to closures.

on a compact subset of R^n, a continuous function is almost Lipschitz: uniform continuity handles small distances with epsilon, while a bound M controls larger distances.