An extension of the sandwich theorem for two-sided limits

Abstract

The criterion for the classical Sandwich theorem for two-sided limits is the existence of two-sided limits for the bounding functions. We show that this criterion can be relaxed. We prove that it is sufficient for the existence of the left-hand limit for the lower bound function and the existence of right-hand limit for the upper bound function; and of-course they must be equal. This paper relaxes the criterion of the Sandwich theorem for two-sided limits, by replacing the two-sided limits with one-sided limits in the criterion and thus, gives an extension of the Sandwich theorem. While Rudin [1] has given a proof of the Sandwich theorem for two sided limits and many has formulated the Sandwich theorem for the one-sided limits, these still don’t relax the criterion of the Sandwich theorem for two-sided limits [2]. They have incorporated the one-sided limits for the Sandwich theorem for one-sided limits but have not relaxed the condition for the Sandwich theorem for two-sided limits as we have done.