$\bigcap I_\gamma$ unbounded if $I_0 \supset \cdots \supset I_{\gamma}\supset \cdots$ are unbounded

Question

Let be $\kappa$ a regular cardinal and $I_0 \supset \cdots \supset I_{\gamma} \cdots$ are unbounded subsets of $\kappa$ for $\gamma < \lambda <\kappa$ where $\lambda$ is limit. I want to show $\bigcap I_\gamma$ is unbounded. Is it true? It's easy show it's not empty for regularity of $\kappa$.

Answer 1

It is false.

Let $\kappa=\omega_1$ and $\lambda=\omega$.

Define $I_0=\omega_1\setminus\{\xi\in\omega_1\mid\xi\mbox{ is a limit}\}$ to be $I_{i}=I_{i-1}\setminus\{\xi+i\in\omega_1\mid\xi\mbox{ is a limit}\}$.

$I_k$ is unbounded but $\bigcap I_i=\emptyset$ as any ordinal can be written as $\mu+n$ where $\mu$ is limit and $n\in\omega$