Is it true that it is impossible to have a rising price level with rising unemployment? If so, Why?

Question

I also have another question: Is it true that our economy is able to adjust to a long-run equilibrium after a decrease in aggregate demand because prices and wages are sticky? If so why?

Answer 1

It is not true. During the 1970's the US hat a period if inflation and rising unemployment. It was called stagflation.
http://en.wikipedia.org/wiki/Stagflation

Answer 2

A rising unemployment level means that fewer people have money, and the people who have savings will be very careful about what they do spend money on. This constriction in the money supply will cause prices to fall, as demand for goods/services will fall.