A Creeping Lemma

Integration of Guided Discovery in the Teaching of Real Analysis

“…One of the topics from the take-home part of the final exam consisted of exploring the following unifying principle [9] of the analysis on the real line and its consequences: Twelve students chose this topic, and all proved the lemma: six used in their argument the compactness of the interval [a, b], and six showed that b is the supremum of the set A = {x E [a, b] | a r x}, one student giving both proofs. 2 When asked to use the result to prove Cantor's Nested Intervals Lemma and the Intermediate Value Theorem, the students were able to find the right transitive relation in each case, but all failed to check the crucial local condition -the existence of the neighborhood N x .…”

“…Four of the questions from my survey are given below. After each question I offer a 2 As explained in [9], the Fundamental Lemma can be used to prove the compactness of [a, b], that is, every open cover has a finite subcover. My students were probably too excited by the topology they learned this term to miss the chance of using the compactness of [a, b] to show the Fundamental Lemma!…”

An Extension of Two Basic Results in Real Analysis

“…(0; 1): See [1], page 11. This result is equivalent to many other basic results such as the Fundamental Lemma of Analysis on R (see [8]). In what follows we shall need a slightly more general version of Cousin's Lemma: The original result of Cousin corresponds to the case where A is the family of all nondegenerate intervals [x 0 ;…”

Using Continuity Induction

“…These were developed by Duren [1] and later by Shanahan [5], [6]. An independent effort, quite close to the present work, is the paper [4] by Moss and Roberts. The most recent incarnation is the paper [3] by Kalantari, who collected these references and formulated continuity induction using conditions similar to (1) and (2).…”