There are two ways to answer this question. One focuses on the mathematical reason and one focuses on the logic that explains why the multiplier cannot be less than 1.
First, let us look at the equation for the simple government spending multiplier. The formula for this multiplier is
Multiplier = 1/(1-MPC).
The MPC is the marginal propensity to consume. Given this equation, it is impossible to come up with a value for the multiplier that is less than 1. Let us imagine that people only consume $.01 of every new dollar that they get from the government. In that case:
Multiplier = 1/(1-.01) = 1/.99 = 1.01
Thus, we can see that even the lowest MPC will yield a multiplier that is at least marginally greater than 1.
Second, let us look at the logic of this. The multiplier exists because people take money spent by the government and then spend it themselves. This multiplies the effect of the government spending. It is not possible for people to spend a negative amount of the money that is given to them. Therefore, there will have to be some amount (however small) of a multiplier effect.