Write the negation of the definition of limit of a sequence given in Example.ExampleThe De...

Write the negation of the definition of limit of a sequence given in Example.

Example

The Definition of Limit of a Sequence

The definition of limit of a sequence, studied in calculus, uses both quantifiers ∀ and ∃ and also if-then. We say that the limit of the sequence an as n goes to infinity equals L and write

if, and only if, the values of an become arbitrarily close to L as n gets larger and larger without bound. More precisely, this means that given any positive number ε, we can find an integer N such that whenever n is larger than N, the number an sits between L − ε and L + ε on the number line.

Symbolically:

∀ε > 0, ∃ an integer N such that ∀ integers n, if n > N then L − ε + ε.

Considering the logical complexity of this definition, it is no wonder that many students find it hard to understand.