Abstract: Using the  different  research methods than the probability space, under the condition that the Choquet integrals existed, we studied the almost sure convergence of weighted sums for extended negatively dependent（END）random variable sequences in the sub-linear expectation spaces, and obtained the almost sure convergence theorem, which extended the theorem from the traditional probability space to the sub-linear expectation spaces.

Key words: sub-linear expectation, END random variable sequence, weighted sum, almost sure convergence