Play with the theorems

sin n never converges — bisect its range and watch nested intervals trap a convergent subsequence anyway.

Walk along sin(1/x) toward the segment and watch the distance meter diverge — connected, but no path gets there.

Steer v₂ and watch its shadow on e₁ get subtracted — and see the algorithm genuinely halt when the vectors go dependent.

A cycle with stationary distribution (⅓,⅓,⅓) that never converges to it — until one slider makes it aperiodic.

Slide the starting point: quadratic convergence one moment, an eternal 2-cycle the next. Local convergence, felt.

Sweep a direction and watch Av swing — except on the eigendirections, where it only stretches. Det appears as the area of the image.

Drag a point across the plane and watch the portrait change: saddle, node, spiral, centre — a map instead of a list of rules.

1/(1+x²) is smooth on all of ℝ, yet its series refuses to go past |x| = 1. Watch the wall the poles at ±i put there.

Slide the map on and watch straight lines bow into circles — every one still a circle or a line, angles preserved throughout.

Forty samples, forty intervals, and a count of how many cover the truth — the misconception this kills is worth several marks.

Watch the initial profile split into halves travelling at ±c, with the light cone marking where the string has not moved yet.

Start from a constant and watch successive integrals close on the solution — agreement spreading outward, which is why existence is only local.

Cosets colouring a cyclic group: same size, never overlapping, always tiling it exactly. The picture will not let you build a counterexample.

Three questions answered from one identity — including the bound most candidates get backwards.

A limit that exists but which L'Hôpital never finds — and the hypothesis everyone forgets to check.

Which test to reach for, in what order, and the family where ratio and root both return 1.

A worked sequence, the subsequential limit set, and the inequality that is only ever one-way.

The one axiom that separates ℝ from ℚ, and the three theorems that collapse without it.

Why 1/x^p flips at p = 1 in opposite directions at 0 and at ∞ — and the integral that converges but not absolutely.

The same escaping-mass sequence run past all three theorems — which apply, which do not, and why.

Why Lebesgue measure cannot be defined on every subset of ℝ — built in four steps.

The inclusion reverses between a probability space and the real line. Both directions, with the counterexample for each.

Why ℝ is not a countable union of nowhere dense sets, and how that one fact proves several others.

Two families that look harmless and are not equicontinuous — and exactly where each breaks.

The topologist's sine curve, and why closure preserves connectedness but not path-connectedness.

The triangular block rule, and the tempting formula that fails unless the blocks commute.

The criterion that always settles it, and the two conditions that look sufficient but are not.

How block sizes are forced by the minimal polynomial and the kernel dimensions.

The form that exists over every field — and the divisibility chain that makes it unique.

Which direction the change-of-basis matrix goes, and the similarity relation that follows.

The leading-minors test done right — including the sign pattern for negative definite that catches people out.

A worked implicit derivative, plus the point where the hypothesis fails and the curve really does misbehave.

Removable, pole or essential — decided by the Laurent tail, with the test for each.

Simple poles, higher-order poles and the check that catches arithmetic slips.

Four hypotheses — which collapse the function to a constant and which only make it a polynomial.

A function smooth on all of ℝ whose series stops at |x| = 1 — explained by a pole you cannot see on the real line.

Order divides — but a divisor need not give a subgroup, and A₄ is the reason.

Three quotients identified in one line each — the fastest way to answer 'what is G/N?'

Order, sign and conjugacy all read off one decomposition — plus where the rule changes in Aₙ.

Counting orbits mod p — the argument behind almost every small-group structure result.

The standard argument, run on order 12 and order 15 — including the element count that finishes it.

Counting the groups of a given order, and the invariant that distinguishes them in one step.

One test settles both — worked in ℤ[x], where an ideal can be prime and still not maximal.

The chain is strict — here is the ring that sits in each gap.

Rational roots, Eisenstein, reduction mod p — and the quartic that factors despite having no real roots.

The tower law in practice, plus the subfield rule for finite fields that decides many options.

A degree-4 extension, its group, and the subfield lattice read straight off the subgroup lattice.

A table you can reconstruct — and the operation that destroys Hausdorff.

Compact implies closed is false without it — here is the space where it fails.

T₁ versus T₂ decided by one example, and the finite-space collapse that is a favourite question.

Cofinite, Sorgenfrey and the Cantor set — the three that answer most topology options.

The auxiliary system, two first integrals, and the general solution — worked end to end.

Real eigenvalues, orthogonal eigenfunctions and simplicity — from one integration by parts.

Why explicit Euler needs h < 2/|λ| however accurate you wanted to be.

Trapezium versus Simpson, and Runge's phenomenon in one example.

When F has no y, and when F has no x — each collapses the equation by one order.

Fixed perimeter, maximum area — set up with a Lagrange multiplier and solved.

The factorial in the iterated kernel that makes one type unconditionally well behaved.

A degenerate kernel reduces the whole problem to a finite system — worked.

Central force in polar coordinates — angular momentum falls out of the Lagrangian in one line.

The Legendre transform done once, plus the reason a Hamiltonian system can never have an attractor.

The tail-integral formula, the two inequalities worth memorising, and a distribution with no mean at all.

One characterising property each for Poisson, exponential and normal.

Derive the max, the min and the k-th — with the dependence that catches people out.

The equivalence holds under joint normality and fails without it — with the standard counterexample.

The uniform case, where calculus finds nothing and the answer sits on the boundary.

Sufficient, then complete, then unbiased — the three steps that produce a UMVUE.

BLUE without any normality — and what normality actually buys you.

The variance decomposition behind stratification, and the periodicity that ruins the systematic scheme.

Derive each error df by subtraction, and know when blocking costs more than it buys.