How do I say "to live up to your name" in Chinese?
I am not sure if there is such an idiom because I tried to find this phrase in various dictionaries (incl. chengyu), but could not find it, so I would be immensely grateful if someone could help me.

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The closest approximation to "live up to your name" in Chinese is 名副其實 or 名實相符。 These expressions mean, approximately, "the name matches the reality."

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名副其实、名符其实应该是最接近的，意思比较接近的还有名不虚传、表里如一、实至名归。

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身名無忝
身分名聲未受汙辱。

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一言九鼎
一言既出驷马难追