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 , one has

.

(2) Show that a real series with summands is convergent, if and only if, for all increasing sequences of positive terms , one has .

(3) For a real sequence and a permutation of the naturals, let . Prove that if a series with summands converges, then the condition is sufficient for the series with summands to have the same sum.

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

converges to

.

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

(5) Let and define the sequences

.

(i) Show that these sequences are equiconvergent.

(ii) Show that there exist such that

.

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 , has the power of the continuum.

(7) For a real function , define the function , wherever this is finite.

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

b) If is lower semi-continuous, then .

c) Define for all irreducible rationals .

Show that this function is lower semicontinuous, but is not finite.

(8) Let be a basis of as a vector space over .

a) Find the cardinality of .

b) Consider a bijection , where is a proper subset of . For all , define .

Show that

(i) .

(ii) is everywhere dense, .

(iii) are unbounded on every interval.

....