Informally, I would like to find an infinite product of rational numbers that evaluates to a nonzero rational number such that the multiplicity of each prime in the numerator is finite, while on the denominator there are an infinite number of primes with unbounded multiplicity.
Existence of an infinite product that converges to a rational number ...
sequences and series - What is the sum of an infinite resistor ladder ...
I am a little confused about how a cyclic group can be infinite. To provide an example, look at $\langle 1\rangle$ under the binary operation of addition. You can never make any negative numbers with
2 Infinite numbers do exist in the hyperreal number system which properly extends the real number system, but then their reciprocals are infinitesimals rather than zero. Thus the idea of $\frac {1} {0}$ can be interpreted as saying that if $\epsilon$ is infinitesimal then $\frac {1} {\epsilon}$ is infinite.
Before what follows, Cantor's diagonal argument was presented as a proof that $\mathbb {R}$ is uncountably infinite; this proof I found to be logically sound. However, after that, an alternative pro...
If the vector space is finite dimensional, then it is a countable set; but there are infinite-dimensional vector spaces over $\mathbb {Q}$ that are countable as sets.
I would like to have some examples of infinite dimensional vector spaces that help me to break my habit of thinking of $\mathbb{R}^n$ when thinking about vector spaces.
Can you partition an infinite set, into an infinite number of infinite sets?