$\aleph_0 \aleph_1 =\aleph_1$? But I don’t know any way to prove or disprove it

Question

What is the value of $\aleph_0 \aleph_1$?

Clearly $\aleph_0\le \aleph_1$ implies $\aleph_0=\aleph_0\aleph_0\le \aleph_1 \aleph_0$ and again $\aleph_0 \aleph_1\le \aleph_1 \aleph_1=\aleph_1$.

But there is no cardinal number between $\aleph_0$ and $\aleph_1$.

So either $\aleph_0 \aleph_1=\aleph_0$ or $\aleph_0 \aleph_1=\aleph_1$.

Then....

Answer 1

HINT: If $\alpha$ and $\beta$ are infinite cardinals, then $\alpha,\beta\leq\alpha+\beta\leq\alpha\cdot\beta$.

Answer 2

Hint: $1\le\aleph_0$, so $1\aleph_1\le\aleph_0\aleph_1$