TITLE: cardinality of all real sequences
QUESTION [28 upvotes]: I was wondering what the cardinality of the set of all real sequences is. A random search through this site says that it is equal to the cardinality of the real numbers. This is very surprising to me, since the cardinality of all rational sequences is the same as the cardinality of reals, and it seemed fairly intuitive to me that if cardinality of a set $A$ is strictly greater than the cardinality of the set $B$, then cardinality of $A^{\mathbb{N}}$ should be strictly greater than cardinality of $B^{\mathbb{N}}$. It turns out to be false.
Some technical answers have appeared in this forum elsewhere but I do not understand them. As I am not an expert in this topic, could some one explain me in simple terms why this is happening?
Also is the cardinality of all functions from reals to reals also the same as the cardinality of reals?

REPLY [4 votes]: Let $k,m$ be infinite cardinals. Then
$$(2^k)^m = 2^{k \cdot m} = 2^{\max \{ k,m \}}.$$
Here I have used the axiom of choice in the second equality. The $\max$ there causes the non-strict monotonicity that surprised you.