True or False?
The statement is false. The problem with this argument, which economists have been known to put forward on occasion, is that unions cannot permanently raise real wages by this tactic so it does not pay them to pursue it. They will bargain instead over real wage levels (incorporating cost-of-living clauses), in which case real wages will rise regardless of the monetary policy of the authorities. At this point, the authorities will not be able to successfully reduce the unemployment rate by committing themselves to keeping it below a level consistent with zero inflation at whatever wage-setting tactics unions adopt. They will thus have no incentive to inflate the money supply.
People who make the argument postulated in the question have in mind a sequential process where the unions raise nominal wages at a given price level, thereby increasing the unemployment rate, then the central bank increases the money supply to raise the price level, eliminating the rise in real wages along with the decline in employment, then the unions again raise nominal wages in order to again increase real wages, then the central bank again increases the money supply.....and so on. The problem is that it does not pay either the unions or the government to play this game once they know what is happening. A smart union will bargain over real wages, at which point the authorities will not be able to eliminate any resulting unemployment by inflating money and prices. This takes away the government's incentive to pursue an inflationary policy.