I understand deltas and epsilon proofs for the most part

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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so 0 < l x-a l < delta and l f(x)-L l < epsilon


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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  • #2
Convention? One less line to draw!
 
  • #3
Reptar said:
so 0 < l x-a l < delta and l f(x)-L l < epsilon


What I don't understand is how come deltas and epsilons can't be greater than or equal to their respective differences?
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].
 
  • #4
Some writers (though not many) even follow the convention that < means "less than or equal to", and reserve [tex]\lneq[/tex] for strict inequality. More commonly, [tex]\subset,\subseteq[/tex] are used interchangeably for set inclusion, with proper inclusion indicated by [tex]\subsetneq[/tex].

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
Dang why is that a good convention? Maybe that just looks weird to me. Anyways I'm perfectly content with just knowing that when I pass off to limits, then I need the non-strict inequality sign. These are rather nitpicky things that I cared way too much about when I started learning analysis. Also, I don't understand the point of getting a nice "less than epsilon" end to an argument, though I admit sometimes it's maybe worth the few extra minutes to finish with "< e" instead of "< e(some ugly factor)".
 

What is a delta and epsilon proof?

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.

Why are delta and epsilon proofs important?

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.

What are the key components of a delta and epsilon proof?

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.

How can I improve my understanding of delta and epsilon proofs?

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.

Are there any common mistakes to avoid when using delta and epsilon proofs?

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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