97 problems.
Day 1: UNL Math 825/826, May 26, 2016 (Avery Hall 347)
Day 2 Metric Spaces and Topology
- Compactness via open covers
- Compactness of the reciprocal-integer set
- Uncountability of the set of binary sequences (warm-up)
- The parallelogram law for inner-product norms (warm-up)
- Open, closed, and compact status of two metric-space subsets (warm-up)
- The sum metric \(d_M+d_N\) on \(M\times N\), and compactness of a product of compact sets
- Infinite unions of compact sets and attainment of distance to a compact set
- Consider the metric space \((\mathbb{Q},d)\)
- A compact set and a disjoint closed set in a metric space are positively separated
- \(\{x:d(x,x_0)=r\}\) is nonempty in an unbounded connected metric space
- A monotone increasing function has at most countably many jump discontinuities
- Open subsets of \(\mathbb R\) are finite or countable unions of disjoint open intervals
- A norm induces a metric \(d(x,y)=\|x-y\|\), but not every metric on a vector space arises from a norm
Day 3 Sequences and Series
- Uniqueness of the limit of a real sequence (warm-up)
- A limit point is the limit of a sequence from the set (warm-up)
- Conditional convergence with divergent square series prevents absolute convergence (warm-up)
- Select a convergent subseries from a sequence tending to zero (warm-up)
- Analyze the Babylonian square-root iteration (warm-up)
- \(\sqrt{n}(\sqrt{n+1}-\sqrt{n})\to\frac12\)
- Produce sequences \(\{a_n\},\,\{b_n\}\) of
- Convergence of \(\sum_{n=1}^\infty \frac{x^n}{1+n|x|^n}\)
- If the series \(\sum_{n=0}^\infty a_n\) converges
- A divergent series of positive terms with \(a_n\to 0\) has a subseries summing to any \(x>0\)
- \(\sum\frac{a_n}{\beta+a_n}\) diverges whenever \(\sum a_n\) diverges, for \(a_n>0\) and \(\beta>0\)
- A bounded positive multiplier preserves convergence of a positive series
- The ratio test follows from the root test
- Use an epsilon–delta proof for a quotient of convergent sequences
- Monotone convergence, the nth-term test, and Cauchy condensation
- Interchange two nonnegative double-series summations
Day 4 Continuity of Functions
- Uniform continuity and a continuous non-uniformly-continuous example
- A continuous self-map of an interval has a fixed point
- An epsilon-delta proof for the cube-root function
- A limit at negative infinity
- A liminf-limsup bound for real sequences
- Uniform continuity of the logarithm on \((0,1]\)
- Extreme value theorem in one real dimension
- Continuity and uniform continuity of a rational function
- Continuity of a running supremum
- A continuous real function cannot take every value exactly twice
- A contraction image containing a bounded set
- A sequential liminf condition implies continuity at a point
- Uniform continuity on compact sets
- Intermediate Value Theorem
Day 5 Differential Calculus
- A function with zero derivative is constant
- A quadratic difference bound forces constancy
- A nonzero derivative gives local injectivity
- The derivative as a limit of difference quotients
- \(\lim_{x\to a}\frac{a^n f(x)-x^n f(a)}{x-a}=a^n f'(a)-na^{n-1}f(a)\)
- Differentiable \(f\le g\) with \(f(x_0)=g(x_0)\) satisfy \(f'(x_0)=g'(x_0)\)
- A bounded derivative on \((a,b]\) implies \(\lim_{x\to a^+}f(x)\) exists
- \(\lim_{x\to\infty}f'(x)=0\) whenever \(\lim f\) and \(\lim f'\) exist at infinity; a counterexample when \(\lim f'\) fails to exist
- \(f:\mathbb{R}\to\mathbb{R}\) with \(f(0)=0\) is differentiable at \(0\) if and only if \(f(x)=xg(x)\) for a function \(g\) continuous at \(0\)
- A differential inequality with zero initial value
- A function vanishing at both endpoints has a large derivative
- A two-sided sequence quotient converges to the derivative
- Fermat’s stationary-point theorem
Day 6 Integral Calculus
- Products of bounded Riemann-integrable functions
- Absolute-value Stieltjes integrability without function integrability
- Epsilon-delta continuity and differentiability of an integral of \(1/t\)
- Vanishing Stieltjes integrals force a monotone integrator to be constant
- A continuous nonnegative \(f\) on \([0,1]\) with \(f(0)>0\) has \(\int_0^1 f>0\)
- The Riemann integral from the defining sums
- \(\lim_{p\to\infty}\bigl(\int_a^b f^p\bigr)^{1/p}=\sup f\) for continuous \(f\ge 0\) on \([a,b]\)
- \(\int_0^1 x^4 f(x)\,dx=\frac15 f(\xi)\) for continuous \(f\) on \([0,1]\)
- \(\int_0^1\cos x\frac{x\phi'(x)-\phi(x)+\phi(0)}{x^2}\,dx<\frac{3}{2}\|\phi''\|_\infty\)
- Reduction of a Stieltjes integral to a Riemann integral
Day 7 Sequences and Series of Functions
- Stone–Weierstrass and a polynomial algebra with high-degree terms
- Pointwise and uniform equicontinuity on a compact set
- Uniform limits of continuous functions are continuous
- Pointwise and uniform convergence of \(x^n\) on \([0,1]\)
- Equicontinuity upgrades pointwise convergence to uniform convergence
- \(f(x^n)\) does not converge uniformly on \([0,1]\) when \(f\) is continuous and \(f(0)\neq f(1)\)
- Equicontinuity of \(\bigl\{\frac{x}{1+nx^2}\bigr\}_{n\ge 1}\) on compact intervals
- If \(f:[0,1]\to\mathbb{R}\) is
- Equicontinuity on \([a,b]\) of differentiable functions with \(|f(a)|\le M\) and \(|f'|\le M\)
- A continuous function on \([0,1]\) orthogonal to \(x^n\) for all \(n\ge 99\) vanishes
- \(\lim_{n\to\infty}\int_{1/n}^1 f_n=\int_0^1 f\) for uniform limits of continuous functions on \([0,1]\)
- Polynomial approximation with endpoint factors
- A trigonometric integral limit
- Uniform convergence along a convergent sequence of points
- A series that is uniformly convergent away from zero but not near zero
Day 8 Miscellaneous Topics
- A function of bounded variation is the difference of two increasing functions
- \(1/f\) has bounded variation on \([a,b]\) when \(f\) does and \(|f|\ge c>0\)
- Bounded variation of x sin(1/x) on the unit interval
- A function with absolutely summable spikes on \([0,1]\) has bounded variation
- A closed bounded noncompact subset of a metric space
- Continuity equivalent to lower and upper semicontinuity, and sequential characterization of lower semicontinuity
- Dini’s theorem: monotone continuous functions decreasing to \(0\) on a compact metric space converge uniformly
- Riemann–Stieltjes integral against a jump at \(0\)
- Riemann-Stieltjes integrability and additivity across a subdivision point
- \(\int_a^b f\,d\alpha=\int_a^b f\alpha'\,dx\) when \(f\) is Riemann integrable and \(\alpha\in C^1[a,b]\)
- Radius of convergence of \(\sum (a_n+b_n)x^n\)
- Radius of \(\sum a_n b_n x^n\) at least the product of the radii of \(\sum a_n x^n\) and \(\sum b_n x^n\), strictly in an example
- Radii of convergence for sums and coefficientwise products of power series
- Uniform convergence of \(\sum_{n=0}^\infty x^n 2^{-nx}\) on \([0,B]\) and on \([0,\infty)\)
- \(\sum f_n\) with \(f_n=\frac1n\) on \((2^{-(n+1)},2^{-n}]\) converges uniformly without the Weierstrass \(M\)-test
- Pointwise and locally uniform convergence of \(\sum\frac{x^k}{1+x^k}\) on \([0,1)\), and differentiability of the sum