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In page 290 of book Weak Convergence and Empirical Processes by Aad van der vaart, Jon Wellner, the last two lines say that the second assertion of theorem 3.2.5 can be proved by minor simplification of the preceding argument, I want someone to provide me the detail of the proof of the second assertion of theorem 3.2.5. ("If the displayed conditions are valid for every $\theta$ and $\delta$, then the condition that $\theta_n$ is consistent is unnecessary.") The picture below is the theorem and its proof.

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Martin Sleziak
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XL Deng
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  • I have edited the links in your post so that both linked pictures are displayed. And I have added the sentence which was underlined in the picture below (in the answer which is now deleted). But you should edit the question yourself until you make clear enough what you're actually trying to ask. – Martin Sleziak Jun 08 '23 at 07:05

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