How is the sequence of $(1/10^n)$ boundless?

Question

I’m reading an old textbook that doesn’t use rigorous methods of proof and only relies on intuition. And the author (Augustus De Morgan) has stated that the sequence 0.1, 0.01... decreases without bound. But isn’t 0 a bound for this sequence?

Answer 1

I think the phrase "diminishes without limit" is intended to mean "it keeps getting smaller forever", as opposed to the modern definition of a sequence having a limit.

More intuitively might be if we take the ratio of N to M instead of the ratio of M to N. This sequence then goes 1, 10, 100, 1000, etc. It's more clear that this sequence "increases without limit", or in modern terms "increases and is unbounded".

Answer 2

Since a ratio cannot be negative, and this ratio cannot be zero, then it makes sense to interpret the statement as meaning: the ratio is not bounded away from zero. More intuitive perhaps, but equivalent, is that the inverse of the ratio grows without bound. That captures the unlimited nature of it, I think.