The infinite monkey theorem states that if you have an infinite number of monkeys each hitting keys at random on typewriter keyboards then, with probability 1, one of them will type the complete works of William Shakespeare.
However, while Dedekind-infinite implies your notion even without the Axiom of Choice, your definition does not imply Dedekind-infinite if we do not have the Axiom of Choice at hand: your definition is what is called a "weakly Dedekind-infinite set", and it sits somewhere between Dedekind-infinite and finite; that is, if a set is Dedekind ...
Except for $0$ every element in this sequence has both a next and previous element. However, we have an infinite amount of elements between $0$ and $\omega$, which makes it different from a classical infinite sequence. So what exactly makes an infinite sequence an infinite sequence? Are the examples I gave even infinite sequences?
Infinite decimals are introduced very loosely in secondary education and the subtleties are not always fully grasped until arriving at university. By the way, there is a group of very strict Mathematicians who find it very difficult to accept the manipulation of infinite quantities in any way.
I had a discussion with a friend about the monkey infinite theorem, the theorem says that a monkey typing randomly on a keyboard will almost surely produce any given books (here let's say the bible...
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.
As far as I can tell, the "infinite matrix" representation of a linear operator is not that popular, especially in non-Hilbert contexts. There are many technicalities to address, as Jesko rightfully points out. Anyway, I remember that I have seen some information on this point of view on the book "Basic operator theory" of Gohberg and Goldberg.
One example which is quite natural is the field given by adjoining all roots of unity or all radicals to $\mathbb Q$. But I must say, I don't like this question: It is not that hard to give you any number of infinite algebraic field extensions (take a suitable collection of polynomials and consider the splitting field), but it seems like there is very little to be learned from such an exercise ...
0 There is a way of proving that there is "infinite number of sides in a circle", in the following sense. Infinity was a number to both Kepler and Leibniz who spoke of a circle as an infinite-sided polygon. This point of view is useful in analyzing the properties of the circle, as well as more general curves, in infinitesimal calculus.
