Suppose that |A-B|<ϵfor each ϵ>0 . Prove that A=B HINT:...

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Key Concepts

Epsilon Arguments

An epsilon argument involves reasoning with arbitrarily small positive numbers. In many proofs, statements are shown to hold for every positive ?, which implies that the quantity under consideration must be as close to a limit or required value as desired. This technique is fundamental in analysis for establishing limits, continuity, and equality by illustrating that any deviation, no matter how small, can be negated.

Proof by Contradiction

Proof by contradiction is a logical technique where one assumes the negation of the statement to be proved and then shows that this assumption leads to a logical inconsistency. In the context of demonstrating equality when a certain inequality holds for every positive number, one assumes the two quantities are different, uses the assumed difference to derive a contradiction with the given bound, and thereby concludes that the assumption must be false.

Absolute Value and Distance

The absolute value represents the non-negative distance between two real numbers. When we say that the absolute difference between two numbers is less than every positive number, it suggests that there is no gap between them—i.e., their distance is zero. This intrinsic property of the absolute value is key to showing that the two numbers must, in fact, be equal.

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