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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
This resolves your problem because it shows that $\frac {1} {\epsilon}$ will be positive infinity or infinite infinity depending on the sign of the original infinitesimal, while division by zero is still undefined. This viewpoint helps account for all indeterminate forms as well, such as $\frac {0} {0}$.
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...