Equivalence of Proof Techniques
Rolando N. Paluga · The Technician. · 2024
Several lemmas have been introduced by various authors to be used as tools in the proofs of theorems in real analysis.They come under various names: Cousin's Lemma, Thomson's Lemma, Creeping Lemma, Weak Creeping Lemma, Ford's Lemma, and Shanahan's Lemma.In this paper, they are shown to be equivalent.There are many articles that introduce tools for proving theorems in real analysis.To cite some, we have Cousin's Lemma (see [3] and [6), Thomson's Lemma (see [1] and [2]), the Creeping Lemma and the Weak Creeping Lemma (see [7]), Shanahan's Lemma (see [8), and Ford's Lemma (see [5).These lemmas have similar applications.For example, using any of these lemmas, we can prove the theorem: If f is continuous on [a,b], then f is Riemann integrable on a,b]. It is not surprising that these lemmas have similar applications since they are equivalent.It is the purpose of this paper to present a proof of the said equivalence.Let us consider the lemmas and the definitions used.A partition of an interval [a,b] is a finite collection of non-overlapping closed intervals whose union is [a,b].A tagged partitio of [a,b] is a partition with one point, referred to as a tag, chosen from each sub-interval comprising the partition.A tagged partition of [a,b] will be denoted by {(c;[x, 1):1 sisn) Where a = Xo < x << xy-< n = b and c; E Xi-, X] is the tag of the interval [x-l for each i.Now let S be a positive function defined on [a,b].A Cousin's Lemma (CL), IfÑ is a positive function defined on the inter-val \a,6], then there exists a &-fine tagged partition of la,6|.For a proof of this, see [3] or (6].KOLANDO N. PALUGA.Professor of