6 Show that if a $\sigma$-algebra is infinite, that it contains a countably infinite collection of disjoint subsets. An immediate consequence is that the $\sigma$-algebra is uncountable.

The reason for this has to to with power series, because the Taylor series is a power series, as well as our approximations. See, if we were to carry out our approximation over and over (in infinite amount …

Jan 27, 2014 · Infinite series of nth root of n factorial Ask Question Asked 11 years, 10 months ago Modified 5 years, 9 months ago

De Morgan's law on infinite unions and intersections Ask Question Asked 14 years, 7 months ago Modified 4 years, 11 months ago

I am looking for a small treatment of basic theorems about infinite products ; surprisingly enough they are nowhere to be found after googling a little. The reason for this is that I am beginning t...

You are right to be suspicious. We usually define an infinite sum by taking the limit of the partial sums. So $$1+2+3+4+5+\dots $$ would be what we get as the limit of the partial sums $$1$$ $$1+2$$ …

Apr 24, 2013 · What might be interesting to notice (although it might be a little too advanced, if this is your first course in topology) that the family of sets in your example is locally finite. Union of a locally …

Jan 14, 2019 · To answer the more general question of when a matrix has no solution: A system of equations can have one of three things: a unique solution, infinitely many solutions, and no solution. …

Feb 9, 2020 · It seems natural that the infinite matrix should also have determinant equal to $1$ but I don't see how the above formula gets this. What about a triangular matrix with diagonal elements …

If determinant is zero, then apart from trivial solution there will be infinite number of other, non-trivial, solutions.