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Suppose $Z_n$ is a sequence of random variables that converge to standard normal in distribution. I saw the claim that $Z_n^2$ converge to chi-square random variable in distribution. But why is that true? How do you prove this?

The only way that I know we can end up with something converges to chi square in distribution is second order delta method, but that doesn't seems to be relevant here.

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This follows that the fact that if $Z_n\to Z$ in distribution, and $h$ is a continuous function, then $h(Z_n)\to h(Z)$ in distribution (a hint for the proof of this fact is here https://math.stackexchange.com/questions/1915776/continuous-function-keeps-the-convergence-in-distribution).

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