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Trying to prove the following about ultrapower embeddings $\pi = k \circ \pi* : V → M$. (The background definitions are on page 3 of these notes by Neeman)

For any $\alpha < \kappa$, $f \in^* Cnst_\alpha$ if and only if there exists $\beta < \alpha$ with $f$ ~ $Cnst_\beta$.

The hint on pg. 4 says to use both Łoś's Theorem and the $<\kappa$ completeness of the ultrafilter $U$. I believe I proved the right-to-left direction without using $<\kappa$ completeness, but am stuck on the left-to-right direction and have no intuition about how $<\kappa$ completeness comes into play. Maybe there is a way (using the hypothesis that $\{\gamma : f(\gamma) \in Cnst_\alpha \} \in U$) of defining the set $\{\gamma : f(\gamma) = \beta \}$ as the intersection of $<\kappa$ many sets in $U$. The first part of my question asks for a hint on this.

Secondly, my difficulty is most likely a deeper confusion about how to use the embedding function when moving between equivalent definitions. For example, does the hypothesis that for $\alpha < \kappa$, $f \in^* Cnst_\alpha$ hold in V or in M*? The relation $\in^*$ is a relation on M*, but the things in M* are equivalence classes, not the functions themselves. When proving the result do I need to apply the $\pi$ function when passing between V and M* (or V and M) using Łoś's Theorem?

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    You might want to look in Jech's "Set Theory" or in Kanamori's "The Higher Infinite" to learn the basics of ultrapower constructions. – 2017-01-03
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    $f\in^* Const_a$ holds in $V.$ – 2017-01-04
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    So then what holds in M* must go through the embedding function $\pi^*$ which gives that $[Cnst_f] \in^* [Cnst_{Cnst_\alpha}]$ which means that the set $\{ \gamma : Cnst_f \in^* Cnst_{Cnst_\alpha} \} \in U$. And this doesn't seem to be what I want to be working with for the proof. – 2017-01-04

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