The Epsilon
University mathematics solved completely. Every step, and the reason for every step: what the step is after, why the move is legal, and what would break without it.
Real Analysis
Real AnalysisLimits, continuity and integrals proved from the definitions, with every inequality justified.| 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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What makes this different
Four promises, kept by software rather than by good intentions.
- Why first
- Every step opens by saying what it is after and why this move corners it. A step without a reason cannot be built by our engine at all.
- Checked
- Both sides of every equation and inequality are computed with exact arithmetic before you see them. If a single line is false, the solution is refused, never served.
- Unlimited
- Type in the problem from your own course, or ask for a fresh one at the level you choose. Each gets the same complete treatment.
- From nothing
- Every symbol is introduced, every inequality flip is justified, every cancelled term is named. Nothing is left as an exercise.
More problem types are on the way
Matrices, eigenvalues and differential equations are next. Leave your email and we will tell you when each one arrives: one short message per new type, nothing else.