Understanding the concept of completeness for an ordered field is known to be difficult for many university mathematics students. We hypothesise that the variety of possible axioms of completeness for the set of real numbers is one of the sources of difficulties as is the lack of understanding of the "raison d'être" of these axioms. In this paper, we address this issue by first recalling the discussions on the first proof of the intermediate value theorem by Cauchy (Cours d analyse De L'École Royale Polytechnique, Part 1: Analyse Algébrique. Debure Frères, 1821) as a contribution to the understanding of the role of completeness in real analysis. Then, we discuss a few research papers that involve the intermediate value theorem and put them in perspective with our own approach. In the second part of the paper, we briefly recall Bolzano's dissection method and then analyse four proofs of the Bolzano--Weierstrass theorem for both the set and the sequence versions of the theorem. We then draw on this study to present and motivate the main features of a forthcoming multi-proofs activity that aims to expose students to a small number of axioms of the completeness property and the use of these axioms in proof and proving.