How to Prove $ \Box (A \wedge B) \rightarrow ( \Box A \wedge \Box B) $ in S5

Question

So far I have:

1. $(A \wedge B)\rightarrow (A \wedge B)$ assumption
2. $A \rightarrow (B \rightarrow (A \wedge B)$ from 1 by propositional logic
3. $\Box A \rightarrow \Box (B \rightarrow (A \wedge B)$ from 2 by RM
4. $\Box(B \rightarrow (A \wedge B) \rightarrow (\Box B \rightarrow (\Box A \wedge \Box B)$ by K
5. $\Box A \rightarrow (\Box B \rightarrow (\Box A \wedge \Box B)$ from 3, 4 by propositional logic
6. $(\Box A \wedge \Box B) \rightarrow (\Box A \wedge \Box B)$ from 5 by propositional lobic

I am wondering if I can infer:

$(A \wedge B) \rightarrow (\Box A \wedge \Box B)$

from any/none of this.

I thought maybe I should apply RN earlier, like two lines before line 1:

$(A \wedge B)$ $\Box (A \wedge B)$

And then go from there but then I ran into issues about whether when using RR on line two I can actually split up the terms the $\Box$ operates over.

Thanks for any and all guidance. This is an exercise in the Chellas book, proving M in S5.