Let a, b ∈ R. Suppose that for every ε > 0, we have a < b + 3ε. Prove that a ≤ b

Question

Let a, b ∈ R. Suppose that for every ε > 0, we have a < b + 3ε. Prove that a ≤ b
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Solution 1

Step 1: Assume for contradiction that a > b.

Step 2: Then, a - b > 0. Let's denote a - b by δ. So, δ > 0.

Step 3: Now, choose ε such that ε = δ/3. Note that ε > 0 because δ > 0.

Step 4: According to the given condition, we have a < b + 3ε.

Step 5: Substituting ε = δ/3 into the inequality, we get Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solve study prob

Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solv

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