e-Proofs: Online resources to aid understanding of mathematical proofs

Analysis is difficult to learn and difficult to teach, not least because it is usually one of the first modules in which students must engage with a large number of abstract proofs. Research in mathematics education indicates that students find it difficult to construct proofs [1,2], that they often behave as though their beliefs about the nature of proof were different from those of expert mathematicians [3,4], that there is often a distinction between what they find personally convincing and what they believe is acceptable as a proof [5], and that their ability to distinguish valid from invalid proofs is unreliable though may improve with ongoing mathematical education [6]. Further, that undergraduates are often unaware of the status of definitions within mathematics [7], which will obviously affect their ability to recognise the use of these within proofs. In real analysis this last difficulty is compounded by the centrality of the limit concept, which has a logically complex definition involving three nested quantifiers.