If you have baby Rudin (Principles of Math. Analysis), see Theorem 6.6. The idea I'm thinking of is to take a partition that includes the points you listed except near zero where you can control the width of the "last" segment in the domain. Then add points near the finite number of points from your list. Then you can control the width on which the inf is zero. On the remaining partitions (most of the area) the inf. is 1. So the lower sum can get arbitrarily close to 1.