An Increase in Government Spending $\Delta$ G That Is Paid for by an Identical Increase in for by an Identical

An increase in government spending $\Delta$ G that is paid for by an identical increase in taxes increases equilibrium output by $\Delta$ Gm regardless of the mpc. Prove.
The decrease in expenditure due to the tax increase is mpc x $\Delta$ G (some comes out of savings), so the total increase in expenditures is $\Delta$ G - mpc x $\Delta$ G = (1 - mpc) $\Delta$ G. The total change in equilibrium output is m(1 - mpc) $\Delta$ G where m is the multiplier, but m = 1/(1 - mpc) so the total change is $\Delta$ G.