I have to find the cardinality of the set of all finite sequences of real numbers.
My solution:
This cardinality is bounded below by the cardinality of the set of real-valued sequences of length one. There are continuum-many such sequences. Now, we need to find the upper bound. The only thing that I can come up with is this:
$$|\bigcup _{n \in \mathbb N}\mathbb R ^n| = \aleph_o \cdot \mathfrak{c} = \mathfrak c$$
And now I employ the Cantor-Bernstein theorem to conclude that the cardinality in question is the continuum.
Do you think that my solution is correct?