Logic Fallacy in a single government purchase creates infinite GDP? Suppose the government increase its spending by $100 million. And suppose we are talking about intermediate level macroeconomic class. Let the marginal propensity to spend be 0.5. 

Now the total GDP should increase by 100/0.5=200 because the increased spending increases people's income and thus further increase the spending, so on and so forth. But wait, what if the marginal propensity to spend is 0! Then we have an infinite increase in GDP! OR if this sounds just wrong, let us consider where the unspent money get saved in banks. And banks will lend the money out which induce firm to invest more. So the investment will increase people's income, and hence the same effect as before. So in the end, we see that the increase in government spending will increase income by the same amount, and then increase spending again by the same amount (the saving also get invested, so it increases the GDP), and etc.

This is clearly not the case in the world we live in. What is the catch?
It seems that you misunderstood the concept of marginal propensity to consume. The marginal propensity is only valid for an infinitesimal increase in income. As soon as you move away from the original starting point where the marginal propensity was 1, there may be a different marginal propensity. The multiplier however assumes that the marginal propensity to consume remains the same at all income levels (since it is essentially an infinitely repeated application of the marginal propensity to consume). 

The above problem with such analyses is why economists switched to models with so-called microfoundations in which the firm and consumer behavior is explicitly modeled via profit/utility maximization. In a model with fixed resources it would quickly become clear that a marginal propensity to consume of 1 for all income levels would violate market clearing conditions.