How to show the number of divergences in quantum gravity is infinite? As is well known, Quantum Gravity is not renormalisable. How can I prove that for the Gravity tensor, there exists an infinite number of divergences? And why can this not be absorbed by mass or gauge transformation?
 A: You cannot really prove, from first principles, that there is an infinite number of divergent amplitudes. The fact that the Einstein-Hilbert Lagrangian contains interactions with negative mass dimensions implies that there exists an infinite number of amplitudes whose superficial degree of divergence is positive.  But this doesn't automatically imply that such diagrams are divergent, because there could be some fortuitous cancellation of terms (usually related to some symmetry) that leads to convergent amplitudes for all but a finite number of diagrams.
In principle, the naïve expectation is that there is an infinite number of divergences, but we have to check this explicitly; after all, the gauge invariance of General Relativity might come to rescue and render the all diagrams finite. As of today, we don't really know if this is indeed what happens or not. However, as far as I know, most people believe that this is not the case: we don't expect that such cancellation occurs. In other words, most people agree that the Einstein-Hilbert Lagrangian is not renormalisable (for more details, see Why is Einstein gravity not renormalizable at two loops or more?).
You might also want to read What's the difference between divergences that can be corrected and those that can't.
