For causal linear time-invariant systems the degree of denominator must be larger or equal than the degree of the denominator.
How can this be proven rigorously?
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I tried to say this in my previous comment: by enforcing causality, you imply realizability thus, there can be no improper transfer function. And the "rigurous" proof:
OTOH, if you want to keep it theoretical, you can even have \$s^n\$ and nobody can stop you, not even the paper you're writing on. You can have as many Diracs, all null before \$t=0\$, and everything will work. But only on paper. Otherwise acceleration could easily be determined by a double differentiator but, in practice, not imposing a bandwidth will bring havok. So the real reason for the transfer function being proper is reality.