I've been fiddling with this enough. Found an answer here but didn't quite understand it.
How do I prove that: $\aleph_0 \cdot \frak{c} \leq \frak{c} \cdot \frak{c}$ $\frak{c} \cdot \frak{c} \leq \aleph_0 \cdot \frak{c}$
The first inequality I've proven already, the second one I just cannot get it(obvious is it may be).
The only thing I'm allowed to use is the definition of cardinality, the cantor-bernstein-schroder theorem and the fact that: $ k \leq p \wedge m \leq n \to k\cdot m \leq p \cdot n $ (which is helpful through the definition of cardinality)
Any insights are welcomed!