Math Analysis 1
The Bolzano-Weierstrass theorem: statement and proof
Every bounded sequence has a convergent subsequence: proof by bisection, common mistakes, examples and what the theorem is used for in the…
Archive
Month: December 2024
6 articles
Math Analysis 1
The intermediate zero theorem (Bolzano): statement and proof by bisection
A continuous function that changes sign on a closed interval has a zero: proof by bisection, a numerical example, applications and what…
Math Analysis 1
The geometric series: when it converges and what it adds up to
Closed form of the partial sum, convergence for |r| less than 1, divergence and oscillation, examples such as 0.999... = 1, common…
Math Analysis 1
The product rule for derivatives: proof and examples
Why the derivative of a product is not the product of the derivatives: an intuition with rectangles, the proof, examples, the quotient…
Math Analysis 1
Limit of a composite function: theorem, proof and change of variable
When you can move a limit inside a function: statement, epsilon-delta proof, a counterexample, change of variable and worked examples.
Math Analysis 1
The limit of sin x / x as x → 0: geometric proof and consequences
Why sin x / x tends to 1 as x tends to 0: proof with areas on the unit circle and the…