with implies that n1 Then S \u2115 Let n2 This is true that 4 4 therefore n is the

# With implies that n1 then s ℕ let n2 this is true

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1.? ∈ 𝑆 2.? ∈ 𝑆with 𝑘 ≥ ? implies that n+1∈ 𝑆 4) Prove if x≥ 0and x≤ 𝜀for each𝜀 > 0, then x =0 Suppose 𝜀, ? ∈ ℝ, by O2, if 0≤ ? 𝑎?? x≤ 𝜀, then 0≤ 𝜀. But since 0 is strictly less than 𝜀 6) a. ||x|-|y|| |x-y| -(x-y) |x|-|y| x-y |x|-|y| x-y |x|-x |y|-y if x ≥ 0 |
b. |x-y| < ? |x|-|y| < |x-y| < c Implies that |x|-|y| < ? and |x| < c+ |y| c. If |x-y| < 𝜀 and 0 < 𝜀 then |x-y|=0 x=y 7) Prove that |x 1 +x 2 + .x n | |x 1| +|x 2| + .|x n | ??? ???? ? ∈ Step 1:|x 1 | |x 1 | Step 2: Assume that |x 1 +x 2 + .x n | |x 1| +|x 2| + .|x n | ??? ???? ? ∈ Step 3: Then |x 1 +x 2 + .x n +x n+1 | |x 1| +|x 2| + .|x n |+|x n+1 |. Suppose |x 1| +|x 2| + .|x n | =a Then |a + x n+1 | |x 1| +|x 2| + .|x n |+|x n+1 | By the triangle inequality, this is true Extra Assigned Problems

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