This paper strengthens the Radner turnpike theorem by examining, under a few reasonable additional assumptions, the mode of the convergence of efficient paths to the von Nuemann ray. Continual proximity to the turnpike is established for long-term efficient paths of economic growth except in a common number of initial and final consecutive periods. This result consolidates the Radner theorem and thereby confirms what the originators of the turnpike proposition undoubtedly had in mind.