Based on the idea of bifurcating a limit cycle from a closed orbit of a nonlinear system, this paper presents a mathematical proof of the pseudooscillator analysis developed recently. The result generalizes the main conclusions of the pseudooscillator analysis, and it can also be used to study the problem of highorder Hopf bifurcation, whose stability requires nonlinear terms with order larger than 3. Two illustrative examples are given for demonstration.