Let $\alpha$ be an ordinal and suppose this ordinal has a cofinal subset. Let the cofinality of $\alpha$ be $\beta$. Thus $\beta$ is isomorphic with $B$, cofinal subset of $\alpha$. Then $\beta$ is a cofinal subset of itself. Now, let the cofinality of $\beta$ be $\gamma$.
Here, how to prove that $\alpha$ has a cofinal subset similar to $\gamma$?
I see this is completely different from proving that cofinal subset of cofinal subset of $A$ is a cofinal subset of $A$, since $\beta$ described above is not a cofinal subset of $\alpha$. (Actually if $\alpha ≠ \beta$, then $\beta$ is not a cofinal subset of $\alpha$.)