 # Real Analysis Proofs

Subject Mathematics Real Analysis

## Question

Prove the following:
Let a,b ∈R. Suppose a > 0. Prove that there is some n ∈∞ such that b ∈[−na,na].

## Solution Preview

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Let a, b ∈ R. Suppose a > 0. Prove that there is some n ∈ N such that b ∈ [−na, na].
We know there is n ∈ N, b < na , by Theorem 2.67 (Archmidean Property).
Suppose b ≥ 0 we know n ≥ 1 and a > 0 therefore −na < 0 and −na < b then b ∈ [−na, na].
Suppose b < 0: There are two cases:
Case 1: There is n ∈ N such that −na < b then b ∈ [−na, na].
Case 2: b < −na for all n ∈ N...

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