I've got the following problem: let $m_0<m_1<m_2<\cdots$ be an increasing sequence of cardinals. Prove that the sum $m_0+m_1+m_2+\cdots$ diffiers from $a^{\aleph_0}$ for any cardinal $a$.
If $a$ is a cardinal such that $$ m_0+m_1+m_2+\cdots=a^{\aleph_0} $$ then, by König's theorem, it must exists $j\in \omega$ such that $m_j \geq a$. Therefore, as the sequence is increasing, it follows that $m_i>m_j \geq a$ for all $i>j$. But now I don't know what could be the next step.
Thanks for the help.
Use König's theorem to argue that $\left(\sum m_i\right)^{\aleph_0}>\sum m_i$; then show that for every cardinal $a$, $(a^{\aleph_0})^{\aleph_0}=a^{\aleph_0}$.