Math Problem Statement
Fix ε > 0. Show that if (an)n∈N,(bn)n∈N are eventually ε-close, then (an)n∈N is bounded if and only if (bn)n∈N is also bounded.
Solution
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Math Problem Analysis
Mathematical Concepts
Sequence Convergence
Bounded Sequences
Eventual Closeness
Real Analysis
Formulas
|a_n - b_n| < ε (for n ≥ N)
Boundedness: |a_n| ≤ M_a
Max Bound: M_a = max(M_a', ε + M_b)
Theorems
Boundedness Theorem
Eventual Closeness
Suitable Grade Level
Undergraduate (Real Analysis Level)
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