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Epsilon-Delta Proofs

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Epsilon-Delta () proof is a proof of a formula on limits based on the epsilon-delta definition. Most students often have difficulties to quickly mastering the epsilon-delta definition and how to use it for proofs! Styles in proof writing usually vary greatly, and no two people will ever be expected to write exact same proof but the logical structure of any two or more proofs should be more or less the same.

In an -proof, it is customary to first do some calculations to find the number , since the proof consists of specifying a value for in terms of and showing that the implication in the limit definition holds for this value of . These calculations to find , may not necessarily be part of the proof.

Let's consider two things, before we look for an actual example:

Absolute Value

If is a real number, the absolute value of is the distance from to , written as . We can alternatively define an absolute value of a real number ,


Therefore, if is any real number, we have



Here, we can naturally, think of as the distance from to .

However, the following are two important equivalences involving absolute value:

.

Here, the symbol meas "is equivalent to". In words: is less than units from if and only if the difference is between and if and only if is in the interval ().

Epsilon-Delta Definition of a Limit

Suppose that and are real numbers and is a function defined in an open interval containing , except perhaps at . If for every positive number there exists a positive number (which depends on ) such that

,

then we say that is the limit of as approaches and we write

.

Intuitively, this definition says that, If is a function defined for all values of near , except perhaps at , and if is a real number such that the values of get closer and closer to as the values of are chosen closer and closer to , then we say is the limit of as gets closer and closer to and we write

.

Example

Show that

Proof

Let and define . Suppose . Then, we have

(since we have made an assumption that and ).

Therefore we have shown that implies ,

and by definition

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