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Ans: I know how to prove hermitian operators eigenvalues are real:

A ket(n) = an ket(n)

bra(n) A ket(n) = an bra(n) ket(n) = an

[bra(n) A ket(n)]* = an* bra(n) ket(n) = an*

[bra(n) A(dager) ket(n)] =[bra(n) A ket(n)] = an*

therefore, an*=an

But, I don't know how to treat an operator^2 and start this question.