Choose a kind of problem

16 kinds of problem in 4 subjects. Each takes your own numbers or invents fresh ones, and every step comes with its reason. The first example of every kind is free to read in full.

Real Analysis

SN	Topic	Problem type	What it solves	Actions
1	Limits from the definition	Limit of a sequence from the definition	Prove that a sequence of the form (an + b) / (cn + d) converges to its limit, using only the definition of convergence: for every tolerance epsilon, find the position M after which every term is within epsilon of the limit.	Open
2	Limits from the definition	Limit of a function from the epsilon-delta definition	Prove that a polynomial (a straight line, a quadratic or a cubic) has the limit you expect at a point, using only the epsilon-delta definition: for every output tolerance epsilon, find an input tolerance delta that guarantees it.	Open
3	The integral, built from nothing	Integral from the definition with upper and lower sums	Show from the definition that an increasing polynomial is Riemann integrable on an interval, and compute the integral exactly by trapping the area between lower and upper sums over n equal pieces.	Open
4	Proof by induction	Proof by induction for a sum	Find the closed formula for a sum such as 1 + 4 + 9 + ... + n squared, and prove it for every natural number n by mathematical induction, with every expansion in the inductive step written out.	Open

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