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The approximation theorem (detailed proof) #ALNT-LB 1.3.DP Chapter 1 Section 3 #Algebraic Number Theory # Lecture note Benois
Theorem 3.1
Let ||-||_1, ..., ||-||_n be inequivalent norms on K, and let a_1, ..., a_n be arbitrary elements of K. Then for any epsilon>0, there exists a in K such that ||a-a_i||_i < epsilon.
Lemma
Let ||-||_1, ..., ||-||_n be inequivalent norms on K, then there exists t in K such that ||t||_1>1 and ||t||_i < 1 for i in {2, ..., n}.
Question:
1. Easy proof of the lemma for n=2 case?
2. A rigorous proof that ||t^N/(1+t^N)|| tends to 1 when ||t||>1 ? (should use triangular inequality)
3. Relation of the theorem with the Chinese remainder theorem?
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By LucTheorem 3.1
Let ||-||_1, ..., ||-||_n be inequivalent norms on K, and let a_1, ..., a_n be arbitrary elements of K. Then for any epsilon>0, there exists a in K such that ||a-a_i||_i < epsilon.
Lemma
Let ||-||_1, ..., ||-||_n be inequivalent norms on K, then there exists t in K such that ||t||_1>1 and ||t||_i < 1 for i in {2, ..., n}.
Question:
1. Easy proof of the lemma for n=2 case?
2. A rigorous proof that ||t^N/(1+t^N)|| tends to 1 when ||t||>1 ? (should use triangular inequality)
3. Relation of the theorem with the Chinese remainder theorem?