Find all field automorphisms of the field of real numbers \( \mathbb{R} \). (A field automorphism of a field \( F \) is a bijective map \( \sigma : F \to F \) that preserves all of \( F \)’s algebraic properties.)

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Find all field automorphisms of the field of real numbers \( \mathbb{R} \). (A field automorphism of a field \( F \) is a bijective map \( \sigma : F \to F \) that preserves all of \( F \)’s algebraic properties.)

Let S be the set of non-zero real numbers x such that there is exactly one 0-1 sequence {a_{n}} satisfying \(\displaystyle \sum_{n=1}^\infty a_n x^{-n}=1\). Prove that there is a one-to-one function from the set of all real numbers to S.