Why is the long-run Phillips Curve vertical?

The Phillips Curve depicts the relationship between unemployment and inflation. Suppose the government pursues an expansionary policy (e.g. lower interest rates). Using the classical model of aggregate demand and supply, we can see that an increase in aggregate demand will result in a fall in unemployment and a rise in inflation (as shown by the Short Run Phillips Curve a.k.a SRPC). However, according to this theory, such a fall in unemployment is only temporary, since workers will begin to expect further price rises in the future and so will demand higher wages. Looking back at the classical model, this will result in a leftward shift in the short-run aggregate supply curve, resulting in a return to the initial level of unemployment but at a higher price level. This is shown by a rightward shift in the SRPC. Therefore, we can say that in the long-run, the Phillips Curve will be vertical because irrespective of the price level, unemployment will return to its natural rate (Natural Rate of Unemployment a.k.a NRU).The Natural Rate of Unemployment is considered the 'sustainable' rate of unemployment because it is composed of supply-side factors (frictional and structural unemployment) rather than demand-side factors. The Long-Run Phillips Curve can therefore only be shifted through supply-side policies (or shocks!).