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.
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...
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?
For many infinite-dimensional vector spaces of interest we don't care about describing a basis anyway; they often come with a topology and we can therefore get a lot out of studying dense subspaces, some of which, again, have easily describable bases.
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 ...
I doubt an infinite number of monkeys could even put together a full page full of nonsense but reasonable-length words with punctuation. You could ask the same question about spiders. Put an infinite number of spiders on typewriters and they won't produce Hamlet either, mostly because most spiders lack the strength to type.
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.
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.
