Infinite set mapping to infinitesimal interval

I would be extremely grateful for any help in understanding this question concerning infinite sets.

Take S to be the set of real numbers from 0 to 1000.

Presumably there is a 1 to 1 function mapping S onto the set of real numbers from 0 to 10, and I expect f(x) = x/100 to be one such function.

But suppose the function was f(x) = x/n, where n is arbitrarily large?

Surely this means that the infinite set S can be mapped on to any real interval, even of infinitesimal size?

In the limit as n à ∞, would that mean that S can be mapped onto any single real number? But then it would no longer be a 1 to 1 function, but ∞ à 1?

Going the other way, the set S could have been mapped onto ever ‘wider’ subsets of the Reals, 0 to 1,000,000,000, then 0 to a googolplex, eventually 0 à ∞, or -∞,à ∞?

Ultimately, couldn’t the entire infinite set of Reals be mapped onto an infinitesimal interval at any single chosen real number? What would happen in the limit as the interval tends to 0? Would it be as though any real number somehow ‘contains’ all real numbers? As though a point can give rise to infinity, and infinity can collapse into a point? [Would similar logic hold for complex numbers?]

If you can sort out my understanding, perhaps by pointing me in the direction of a few articles for me to read, I would be very grateful.

Thank you very much for your time.

Infinite set mapping to infinitesimal interval

Thank you for your very swift replies, CaptainBlack and Math Major. Things are getting much clearer.

The set of Reals can be mapped to an interval, as long as it is finite.

There are bijective mappings, but this would not be needed if all that was required was to cover all Reals in the domain?

A finite interval, no matter how small, can be mapped to the set of Reals.

This mapping could be bijective, but would at least need to be surjective in order to generate all Reals?

Although the interval cannot be shrunk to a specific 'point'/number on the number line, if there is any uncertainty/probability associated with a number, then an interval would be implicit?

I've just realised that the direction I am taking may well be leading the discussion out of the sub-forum. Apologies.

Thank you for your clarifications.

Fred

What sort of interval/set can be used generate the Reals

Thanks, Math Major, for your detailed explanations.

The Wiki reference was also very interesting.

What most intrigues me is that the unaccountably infinite set of Reals can be 'generated' (by a variety of functions and mappings) from an interval of almost zero 'diameter'/length.

Could we get some sort of lower bound on the 'smallest' size of that interval/set?

Does that almost infinitesimal interval/set have to have the same cardinality as the Reals, aleph1?

Or could it instead be a countable infinite set (aleph0), and a mapping that maps each element in the set to an unaccountably infinite subset of the Reals, such that all the Reals would be covered?

But I suppose there would be no way to guarantee full coverage of the Reals? One would always be able to design a number that could not be generated by the countable infinite set of mappings being used?

Regards

Fred