Advanced Calculus/Real Analysis with the Math Sorcerer

Explore the archimedean property, showing every positive number is surpassed by a natural number, and prove 1/n < epsilon for all n ≥ N.

Apply the archimedean principle to any real x. Find n1 > x and n2 > -x, set n = max(n1, n2), and conclude -n < x < n.

Using the archimedean property, obtain n1 with x < n1 and n2 with 1/n2 < x. Set n = max(n1, n2) to ensure 1/n < x < n.

Demonstrate the uniqueness of a sequence’s limit by assuming two different limits and using an epsilon argument with the triangle inequality to reach a contradiction.

The lecture presents a rigorous epsilon-based convergence proof that a sequence converges to one, using Archimedes' principle to select N and bound |a_n - 1|.

The lecture proves that the sequence (-1)^n diverges by contradiction, using epsilon equals 1 and the even and odd terms to show no convergence.

Demonstrate, using the limit definition, that if x_n -> 2 then (2 x_n - 1)/3 -> 1 by showing |(2 x_n - 1)/3 - 1| = (2/3)|x_n - 2| and choosing a bound.

Examine a divergent sequence with a convergent subsequence, using the oscillating sequence (-1)^n where the even terms converge to 1, illustrating coexistence of divergence and subsequence convergence.

Demonstrates that every convergent sequence is a Cauchy by epsilon arguments and the triangle inequality: if a_n → L, then |a_n − a_m| < ε for all n, m > N.

Show that the sequence a_n = cos(1/n) is a Koshy (Cauchy) sequence by bounding |cos(1/n) - cos(1/m)| with the cosine identity and the Archimedean property to stay below any epsilon.

Demonstrate a delta-epsilon proof that the limit of 3x+5 as x approaches 2 is 11 by choosing delta equals epsilon over three and outlining how scratch work informs formal proof.

Illustrates a delta-epsilon proof showing lim x→2 of (x^2−4) = 0 by bounding |x−2||x+2| and choosing delta = min(1, epsilon/5).

Explore a delta-epsilon proof that the limit of x cubed as x approaches 2 is 8, using the difference of cubes to bound |x^3-8| by |x-2|(x^2+2x+4) and delta = min(1, epsilon/19).

Demonstrate a delta-epsilon proof that if lim x→a F(x)=L and lim x→a G(x)=K, then lim x→a [F(x)+G(x)]=L+K, using delta as the minimum and epsilon/2 with the triangle inequality.

Apply the squeeze theorem to evaluate the limit as x approaches 1 of (x^2-1)^3 sin(1/(x-1))^3, using sine's bound between negative one and one to conclude the product tends to zero.

Proves the sine function is continuous at every real number using the delta-epsilon definition and trig identities.

Show that the Dirichlet function is nowhere continuous by contradiction, using the epsilon-delta definition and the density of rationals and irrationals.

Explain the difference between continuity and uniform continuity, showing how delta depends on x and epsilon in continuity but only on epsilon in uniform continuity.

Demonstrates that f(x)=x^2 is uniformly continuous on (0,1) by an epsilon-delta proof. Use delta = epsilon/2 and the difference of squares to bound |f(x) - f(y)|.

Prove cosine is uniformly continuous on R using cos x − cos y = −2 sin((x+y)/2) sin((x−y)/2) and bound |cos x − cos y| ≤ |x−y| with delta = epsilon.

Prove sine is uniformly continuous on real numbers. Use sin x − sin y = 2 cos((x+y)/2) sin((x−y)/2) and |sin t| ≤ |t|, |cos t| ≤ 1, delta = epsilon.

Demonstrate the uniform continuity of x cubed on the interval from zero to two using delta–epsilon methods and the difference of cubes with a bound of 12 for x^2+xy+y^2.

Demonstrate that f(x) = x/(x−1) is uniformly continuous on [2, ∞) by choosing δ = ε and showing |f(x) − f(y)| ≤ |x − y| for x, y ≥ 2.

Show that the function f(x) = 1/x on the interval (0,1) is not uniformly continuous by contradiction, using epsilon = 1 and y = x/2.

this lecture proves that a differentiable function is continuous at that point, using the derivative definition to show the limit of f(x) as x approaches c equals f(c).

Prove the derivative of cosine equals negative sign by evaluating the limit of (cos(x+h)-cos x)/h, using cos(x+h)=cos x cos h - sin x sin h, and the limits lim(cos h-1)/h=0 and lim(sin h)/h=1.

Prove that a piecewise function is differentiable for all real numbers by evaluating the limit at zero, using the squeeze theorem to show f'(0)=0, and noting differentiability elsewhere.