Why there is liquidity trap in the reality?

When the interest rate is low enough, monetary policy will fail to work, because:

1. interest rate cannot be lower than zero. Given the circumstances that the interest rate is low enough, the amount of money central bank inject into the private banks will not reduce the interest rate further.

2. consumption will not be boosted by decreasing in interest rate when it is approaching zero. Interest rate is no the only factor that influence the household consumption. When the interest rate is very low, a further decrease brings no difference.