If $cf(\kappa)=\lambda$, then is every sequence of length $\lambda$ cofinal in $\kappa$?

Question

Take $\omega_1$ for instance. Let's say I have a sequence of (distinct) ordinals of length $\omega_1$. Will this sequence be cofinal in $\omega_1$?

Answer 1

The answer to your question about $\omega_1$ is yes, but the answer to the title is in general no.

For $\omega_1$, note that, for each $\alpha<\omega_1$, your sequence of distinct ordinals contains only countably many terms $\leq\alpha$, so, from some point on, your sequence is above $\alpha$. Since $\alpha$ was arbitrary (below $\omega_1$), this means your sequence is cofinal in $\omega_1$.

But for the general situation, consider $\kappa=\aleph_\omega$ (also known as $\omega_\omega$), so $\lambda=\omega$. The identity function on $\omega$ is obviously not cofinal in $\aleph_\omega$.