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Finding a basis of an infinite-dimensional vector space? 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
infinity - What is the definition of an infinite sequence . . . 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?
Proof of infinite monkey theorem. - Mathematics Stack Exchange 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
Infinite Cartesian product of countable sets is uncountable So by contradiction, infinite $0-1$ strings are uncountable Can I use the fact that $\ {0,1\}$ is a subset of any sequence of countable sets $\ {E_n\}_ {n\in\mathbb {N}}$ and say the infinite product of this is uncountable too?
How can Cyclic groups be infinite - Mathematics Stack Exchange 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