Relation between Signal to Noise Ratio (SNR) and Transmission Bandwidth for FM signal
This document entitled Frequency Modulation (FM) Tutorial by Lawrence Der, Ph.D. of Silicon Laboratories Inc. shows that: -



What the above is saying is that if FM deviation doubled (transmission band width doubled), SNR would increase 8 times. See also this from wiki: -



In your question there has to be some form of error because if SNR is "directly proportional to BW\$^2\$" and BW doubles then SNR must rise by 4. 


  How can we prove this?


Basically, you can't prove something that is wrong without lying or some form of deception.