- #1

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What I don't understand is how come deltas and epsilons can't be greater than

__or equal__to their respective differences?

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In summary: So, in summary, the convention in math is to use "less than" instead of "less than or equal to" for deltas and epsilons because it is more concise and still allows for strict inequalities to be used in certain cases. This convention may seem nitpicky, but it is important in analysis when dealing with limits and sequences.

- #1

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What I don't understand is how come deltas and epsilons can't be greater than

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- #2

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Convention? One less line to draw!

- #3

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Because it is "less than" that is important- and if I can find [itex]\delta[/itex] so that [itex]0< |x- a|\le \delta[/itex] I could always choose [itex]\delta[/itex] just slightly larger and have [itex]0< |x- a|<\delta[/itex].Reptar said:

What I don't understand is how come deltas and epsilons can't be greater thanor equalto their respective differences?

- #4

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It's not a bad convention IMO, because, for instance, for a sequence converging to L, you can write [tex]\forall n,a_n<c\Rightarrow L<c[/tex], without worrying that limits don't preserve strict inequality.

- #5

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A delta and epsilon proof is a mathematical technique used to prove the limit of a function. It involves selecting a delta value and an epsilon value to show that for all inputs within delta distance from a given point, the output will be within epsilon distance from the limit point.

Delta and epsilon proofs are important because they provide a rigorous and formal way to prove the existence of limits in calculus. They also help to establish the convergence of sequences and series, which is essential in many areas of mathematics and science.

The key components of a delta and epsilon proof are the definition of a limit, the selection of a delta value, the selection of an epsilon value, and the use of mathematical inequalities to show that the output is within epsilon distance from the limit point for all inputs within delta distance.

To improve your understanding of delta and epsilon proofs, it is important to practice solving problems and to work through proofs step by step. It is also helpful to review the fundamental concepts of limits, continuity, and mathematical inequalities.

Yes, some common mistakes to avoid when using delta and epsilon proofs include selecting an incorrect delta or epsilon value, making incorrect assumptions about the function or limit point, and using incorrect mathematical manipulations. It is important to carefully follow the steps of the proof and to double check your calculations to avoid these mistakes.

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