Cardinality of the set of all infinite monotonically decreasing sequences of naturals

Question

Find the cardinality of the set of all infinite monotonically decreasing sequences of naturals.

I think it's $\aleph_0$. I marked this set in $A$, and said that $\forall n\in\Bbb N \ (n,n,n,...)\in A$, Therefore, $\aleph_0\le|A|$. But I'm not sure how to show that $|A|\le\aleph_o$. Thanks

Answer 1

A monotonically non-increasing sequence of natural numbers is eventually constant, so it is completely identified by specifying the finite initial subsequence up to the point at which it becomes constant. There are only countably many finite sequences of natural numbers.

Answer 2

Find the cardinality of the set of all infinite monotonically decreasing sequences of naturals.

There are no infinite monotonically decreasing sequences of naturals, since the natural numbers are well-ordered. Thus, the set of all such sequences is empty, and has cardinality $0$.

You seem to have gotten decreasing mixed up with non-increasing.