No, he was right. The fact that the mass of the proton mas increases by a factor of 7,500 is fairly amazing. If a 100 pound weakling made the same speed, he (or she) would weigh 375 tons. Granted, that's not "infinite," but it's also not going the speed of light in a vacuum.
The fact that the mass increases is why it takes so much energy to impart more speed, once one gets to relativistic velocities. At this velocity, slight increases in velocity create significant increases in mass.
Said another way, the "TeV" limit results in a mass (and corresponding speed) limit. It would take infinite energy to obtain infinite mass.
OK.
I *THINK* I understand, now.
The mass is increasing dramatically . . . but only because of the dramatic amount of energy used to speed the protons up. And, still, though they are at a higher end of such speed-ups than ever before achieved, the percentages left between their highest and truly the speed of light are still significant distances to cover . . . and would require more or less infinite energy to get there—at least as we currently seem to understand things.
Is that close to right?