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I am reading a proof in Van der Vaart and Wellner's Empirical Process Theory and they state that if a net $X_\alpha$ is asymptotically tight, then so is $g(X_\alpha)$ for any continuous function $g$. I believe are working in arbitrary complete metric spaces.

Asymptotic tightness is defined as: for all $\varepsilon$, there exists a compact set $K$ such that:

$\lim \inf_\alpha P_*(X_\alpha \in K^\delta) \geq 1-\varepsilon$, for all $\delta > 0$.

I'm having trouble showing this and would appreciate any help. I have tried various things, including trying to show that:

(1) $g(\overline{K^\delta})$ is compact

(2) $g(K)^\tilde{\delta}$ contains $g(K^\delta)$ for some $\delta$ small enough

But haven't found a way to make them work. Thanks!

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