Past exam of the martingales and financial mathematics course of the University of Oxford / 2010 / A / 1 / b / i / Solution

The triangle inequality gives the event inclusionCountable subadditivity of a measure bounds its measure by the sum of the two terms on the right. Both tend to zero as $m,n\to \infty$ by convergence in measure. Thus convergence in measure implies the Cauchy property in measure.