Looking through boxes of old books, i ran across a Nicholas Bourbaki volume...

(Don't know if you 've heard the story - take a minute to read. This group of French mathematicians, around 1890, decided to rewrite math from scratch. So they massed volumes over volumes, under the pseudonym "Bourbaki". This guy become famous, and the pranksters even got actors to occasionaly appear as N. Bourbaki and give lectures, without knowing anything about math... )

Here's a little list of problems, collected from "Topologie Generale". It gives me chills to think, that some people had to struggle with these, as part of their courses...

(1) Show that for all integers $\displaystyle p>1$, one has

$\displaystyle \sum_{n}\frac{(-1)^{n-1}}{n^p}=(1-2^{1-p})\sum_n\frac{1}{n^p}$.

(2) Show that a real series with summands $\displaystyle (u_n)$ is convergent, if and only if, for all increasing sequences of positive terms $\displaystyle (p_n)$, one has $\displaystyle \rm{lim}\sum_{0\leq k\leq n}\frac{p_k u_k}{p_n}=0.$.

(3) For a real sequence $\displaystyle (u_n)$ and a permutation $\displaystyle \sigma(n)$ of the naturals, let $\displaystyle r(n)=|\sigma(n)-n|\rm{sup}_{m\geq n}|u_m|$. Prove that if a series with summands $\displaystyle (u_n)$ converges, then the condition $\displaystyle \rm{lim}_nr(n)=0$ is sufficient for the series with summands $\displaystyle (u_{\sigma(n)})$ to have the same sum.

(4) Let $\displaystyle (\epsilon_n)$ be a sequence of terms equal to ±1. Show that the sequence

$\displaystyle \epsilon_1\sqrt{2+\epsilon_2\sqrt{2+\epsilon_3 \sqrt{2+ \ldots+ \epsilon_n\sqrt{2}}}}$

converges to

$\displaystyle 2\sin\left(\frac{\pi}{4}\sum_{k}\frac{\epsilon_1 \ldots \epsilon_k}{2^k}\right)$.

(good lord!!! that's an exam I'd hate to take)

(5) Let $\displaystyle 0< y_0\leq x_0$ and define the sequences

$\displaystyle x_{n+1}=\frac{x_n+y_n}{2}, \ y_{n+1}=\sqrt{x_ny_n}$.

(i) Show that these sequences are equiconvergent.

(ii) Show that there exist $\displaystyle a>0, \ 0<\gamma\leq 1$ such that

$\displaystyle x_n-y_n\leq a\gamma^{2^n}$.

Hint. Consider differences of squares.

(I laughed my heart out, over this precious piece of advice... )

(6) Show that the set of all perfect compact subsets of $\displaystyle \bold R$, has the power of the continuum.

(7) For a real function $\displaystyle f$, define the function $\displaystyle \omega(f;x)=\rm{limsup}_xf-\rm{liminf}_xf$, wherever this is finite.

a) Prove that this function is upper semi-continuous on its domain of definition.

b) If $\displaystyle f$ is lower semi-continuous, then $\displaystyle \rm{liminf}_x\omega=0$.

c) Define $\displaystyle f(r)=q$ for all irreducible rationals $\displaystyle r=\frac{p}{q}$.

Show that this function is lower semicontinuous, but $\displaystyle \omega(r;f)

$ is not finite.

(8) Let $\displaystyle \bold B$ be a basis of $\displaystyle \bold R$ as a vector space over $\displaystyle \bold Q$.

a) Find the cardinality of $\displaystyle \bold B$.

b) Consider a bijection $\displaystyle \phi:\bold C\rightarrow \bold B$, where $\displaystyle C$ is a proper subset of $\displaystyle B$. For all $\displaystyle x=\sum_{\xi\in \bold B}\lambda(\xi)\xi$, define $\displaystyle f(x)=\sum_{\xi\in \bold B}\lambda(\xi)\phi(\xi)$.

Show that

(i)$\displaystyle f(x+y)=f(x)+f(y)$.

(ii) $\displaystyle f^{-1}({z})$ is everywhere dense, $\displaystyle \forall \ z\in \bold R$.

(iii) $\displaystyle \rm{liminf}_xf, \ \rm{limsup}_xf$ are unbounded on every interval.

....