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On the Hegselmann–Krause conjecture in opinion dynamics

We give an elementary proof of a conjecture by Hegselmann and Krause in opinion dynamics, concerning a symmetric bounded confidence interval model: If there is a truth and all individuals take each other seriously by a positive amount bounded away from zero, then all truth seekers will converge to the truth. Here, truth seekers are the individuals who are attracted by the truth by a positive amount. In the absence of truth seekers, it was already shown by Hegselmann and Krause that the opinions of the individuals converge.