I searched but couldn't really find an answer for that, so sorry if its a duplicate or anything else.
My question is why a set, for example A, that has a subset/equal set which is infinite(N for example), isn't infinite by definition? The formal definition for an infinite set is "A set is infinite if and only if for every natural number the set has a subset whose cardinality is that natural number."
It makes sense of course, and its easy to prove that A is also infinite, but then why wouldn't N ⊆ A also mean that A is infinite by definition? Is there an opposite example when it doesn't happen? I'm not very familiar with math definitions in english, so sorry if I wrote something wrong. Thanks.
Edit - @Asaf Karagila answered it perfectly. Thanks for the replies.