equals if the following condition holds: For all , there exists a tagged partition and such that for any tagged partition and which is a refinement of and , we have Both of these mean that eventually, the Riemann sum of with respect to any partition gets trapped close to . Since this is true no matter how close we demand the sums be trapped, we say that the Riemann sums converge to . These definitions are actually a special case of a more general concept, a net. As we stated earlier, these two definitions are equivalent. In other words, works in the first definition if and only if works in the second definition. To show that the first definition implies the second, start with an , and choose a that satisfies the condition. Choose any tagged partition whose mesh is less than . Its Riemann sum is within of , and any refinement of this partition will also have mesh less than , so the Riemann sum of the refinement will also be within of . To show that the second definition implies the first, it is easiest to use the Darboux integral. First, one shows that the second definition is equivalent to the definition of the Darboux integral; for this see the Darboux Integral article. Now we will show that a Darboux integrable function satisfies the first definition. Fix , and choose a partition such that the lower and upper Darboux sums with respect to this partition are within of the value of the Darboux integral. Let If , then is the zero function, which is clearly both Darboux and Riemann integrable with integral zero. Therefore, we will assume that . If , then we choose such that If , then we choose to be less than one. Choose a tagged partition and with mesh smaller than . We must show that the Riemann sum is within of . To see this, choose an interval . If this interval is contained within some , then where and are respectively, the infimum and the supremum of f on . If all intervals had this property, then this would conclude the proof, because each term in the Riemann sum would be bounded by a corresponding term in the Darboux sums, and we chose the Darboux sums to be near . This is the case when , so the proof is