porcupinetree
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Kingofacting has helped me to prove this for real numbers a_n and b_n, but not integers... can someone shed some light?

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Kingofacting has helped me to prove this for real numbers a_n and b_n, but not integers... can someone shed some light?
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What's getting me is that we need to use the n=2 case in the inductive step, but (as far as I see) there's no guarantee that the statement holds true for integers for the n=2 case (like, what if all the entries of M are irrational?)
Ok yeah that makes sense. Thanks
Edit: Sorry, I think what I wrote it wrong (fails for n = 2. But if you can show the n = 2 case, you can certainly induct this way.).
Yeah it won't hold if M can be any real matrix at all (edited in an example).What's getting me is that we need to use the n=2 case in the inductive step, but (as far as I see) there's no guarantee that the statement holds true for integers for the n=2 case (like, what if all the entries of M are irrational?)