Logic – Finite Modal Property via Selection-Question for Bisimulation

I follow the proof of:

To let $ tau $ be a modal similarity type containing only diamonds, and be $ phi $ be a $ tau $-Formula. If $ phi $ is satisfactory, then it is satisfactory on a finite model.

I just prove it for the basic modal language, that is, we only have one diamond $ diamond $

The proof is:

Fix a modal formula $ phi $ With $ deg ( phi) = k $, We limit our modal similarity type $ tau $ and our collection of sentence letters to the modal operators and sentence letters that actually occur in Germany $ phi $, To let $ M_1, w_1 $ be so $ M_1, w_1 vdash phi $
, By Theorem 2.15, a tree-like model exists $ M_2 $ with root $ w_2 $ so that $ M_2, w_2 vdash phi $
, To let $ M_3: = (M_2 upharpoonright k) $, With Lemma 2.33 we have $ M_2, w_2 $ is $ k $-Imilar to $ M_3, w_2 $and by Theorem 2.31 it follows $ M_3, w_2 vdash phi $,

The construction of $ M_4 $ starts here.

By induction on $ n le k $ We define endless sets of states $ S_0, …, S_k $ and a (last) model $ M_4 $ with domain $ S_0 cup cdots cup S_k $; the points in each $ S_n $ will have height $ n $, Define $ S_0 $ to be the singleton $ {w_2 } $, Next, accept this $ S_0, …, S_n $ have already been defined. Fix an item $ v $ from $ S_n $, By Theorem 2.29, there are only infinitely many non-equivalent modal formulas whose degree is at most $ k $, say $ 1, …, m $, For each of these formulas is this form $ diamond phi $ and stops in $ M_3 $ at the $ v $, Choose a state $ u $ from $ M_3 $ so that $ Rvu $ and $ M_3, u vdash phi $, Add all these to us $ S_n + 1 $and repeat this selection process for each status in $ S_n $, $ S_ {n + 1} $ is defined as the set of all points selected in this way.

Finally define $ M_4 $ as follows. Your domain is $ S_0 cup cdots S_k $; like everyone else $ S_i $ is finite, $ M_4 $ is finally. The relationships and assessments are obtained by restricting the relationships and evaluations of M3 to the domain of M4. We can prove that $ M_4, w_2 $ is $ k $-Imilar to $ M_3, w_2 $,

The last sentence is actually an exercise that is not easy enough to solve within a few hours. Could someone please tell me how I can prove it (use some relative phrases?) Or give a hint, please?

Thanks for any help.