TITLE: Can a set be infinite and bounded?
QUESTION [11 upvotes]: I don't understand a statement in my math book course, I was restudying the compact sets part of the chapter when at a certain moment there is a corollary saying :
'every infinite and bounded part of $\mathbb{R^n}$ admit at least one accumulation point'
because for me a set is either bounded so finite or infinite so unbounded.
I don't really understand because I can accept the fact that without a metric, bounds make no sense in topology but here $\mathbb{R^n}$ is clearly known as a metric space.
thank you for your help

REPLY [2 votes]: infinite refers to the cardinal of the set.
bounded refers to the values.
$(1, \frac{1}{2},\frac{1}{3},\frac{1}{4} ...)$ is a set that defines an infinite number of values, all of which are included in the interval $(0, 1]$
It is infinite and bounded.