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2012/11/26

epsilon-delta proofs 4

In [Def-1] we defined a real number sequence a_n approaching a. Although [Def-1] is much complicated, the reason why we should use it is that we were able to discuss precisely.

Using mathematical symbols, [Def-1] is also expressed as follows.

a_n approaches a, when
 (\forall \epsilon>0)(\exists N\in \mathbb{N})(\forall n\in \mathbb{N})(n\geq N\rightarrow |a_n-a|< \epsilon) 
where \forall  means 'all ' or 'any', \exists  means 'exist ' or 'some', and \in  means 'in elements of '.

Make a contrapositive statement of [Def-1] forcibly.  Next sentence is the answer. If
    (\exists \epsilon>0)(\forall N\in \mathbb{N})(\exists n\in \mathbb{N}) (n\geq N \land |a_n-a|\geq \epsilon), 
a real number sequence a_n never approaches a, where \land means 'and '.

That is, a_n does not approach a when for some \epsilon>0 , any N>0, there is n>0 such that n\geq N and |a_n-a|\geq \epsilon .

The sequence 2.3888\cdots does not approach 2.4. But the sequence 2.3888\cdots approaches 43/18. One of the sequence which approaches 2.4 is 2.3999\cdots . These shows that the converging depends on the relation between a sequence a_n and a value a.

We are interested in the case in which a_n does not converge with any values. It is called diverging. In particular, if the greater n becomes, the greater a_n becomes, we say that a_n approaches infinity, or the limit of a_n is \infty and expresses that
 \lim_{n\rightarrow \infty} a_n =\infty, \quad or \quad a_n\to\infty \quad (n\to \infty )
Next definition is equivalent to above.

[Def-3] a_n approaches infinity, when for any \epsilon >0, there is N>0 such that if n\geq N, |a_n-a|\geq \epsilon.

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