 If a sequence of real numbers ("x""i") converges to a real number "x", then by definition, for every real ε > 0 there is a natural number "N" such that if "i" > "N" then |"x" − "x""i"| < ε.A modulus of convergence is essentially a function that, given ε, returns a corresponding value of "N".Suppose that ("x""i") is a convergent sequence of real numbers with limit "x".