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Seacow1988
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Why is it true that: if a+e<b for all e>0 then a≤b? What is the meaning of epsilon here?
Thanks!
Thanks!
Seacow1988 said:Why is it true that: if a+e<b for all e>0 then a≤b?
Seacow1988 said:Let e > 0 be given. Since α−e/2 < α = supA, we can find a in A such
that α−e/2 < a. Similarly, we can find b in B such that β−e/2 < b.
Let c = a + b. Then (α + β) − e = (α − e/2) + (β − e/2) < a + b = c
and c belongs to A+B. It follows that (α+β)−e < sup(A+B) = γ.
Since this holds for all e > 0, if follows that α + β ≤ γ.
… this is: if a-e<b for all e>0 then a≤b.Seacow1988 said:Why is it true that: if a+e<b for all e>0 then a≤b?
The "Basic Epsilon Question" is a fundamental concept in the field of mathematics and physics. It refers to the smallest positive number that can be represented in a given numerical system. This number is often denoted as ε and is used to measure the precision of calculations and to determine the convergence of numerical series.
The "Basic Epsilon Question" is used in various scientific fields, including mathematics, physics, and computer science. It is an essential concept in numerical analysis, where it helps determine the accuracy and convergence of algorithms and calculations. In physics, it is used in theories such as quantum mechanics and general relativity, where it represents the smallest measurable quantity. In computer science, it is used in programming to compare the precision of different data types and to prevent errors in calculations.
The value of epsilon is dependent on the numerical system being used. In the decimal system, the value of epsilon is 0.000...001, where the number of zeros after the decimal point can vary depending on the precision needed. In other numerical systems, such as binary or hexadecimal, the value of epsilon may be different. However, the concept of epsilon remains the same, representing the smallest positive number in that system.
In calculus, epsilon is used to define the limit of a function at a specific point. It is used in the formal definition of a limit to determine how close the input value must be to the limit point in order for the output value to be within a certain range. This concept is crucial in understanding the behavior of functions and is used extensively in advanced calculus and real analysis.
The "Basic Epsilon Question" and the concept of infinity are closely related. In many cases, epsilon is used to represent a value approaching infinity, as it gets closer and closer to being infinitely small. In calculus, for example, epsilon is used to define the limit of a function at infinity. Additionally, epsilon is used in theories such as the theory of limits to prove the existence of infinite sets and to determine their properties.