Let be a sequence of real numbers and . The sequence is said to converge to () iff:
1.1 Cauchy Sequences
Let be a sequence of real numbers. Then, is said to be a Cauchy sequence iff:
Any sequence of real numbers is convergent if and only if it is a Cauchy sequence.
2. Metric Spaces
A metric space is a tuple where is a non empty set, and is a mtric over . Meaning: such that:
.
.
.
.
We can convert a vector space into a metric space. Let be a vector space equipped with the norm . Then we can define a metric space where is defined as:
.
Now, we can generalize concepts of convergence and Cauchy sequences to metric spaces. Let be a metric space and be a sequence in . Then is said to converge to iff:
If is converging, then the limit is unique.
2.1 Cauchy Sequences
Similarly, we can define a Cauchy sequence in a metric space with a sequence . Then is said to be a Cauchy sequence iff:
If is a metric space and is convergent, then it is a Cauchy sequence.
2.2 Complete Metric Spaces
Let be a metric space. Then is said to be complete iff every Cauchy sequence in is convergent in .
For any , is a complete metric spaces equipped with any of the metrics induced by , , or norms.
The space of continuous functions on is a complete metric space equipped with the norm.
If is a complete metric space then is converging iff it is a Cauchy sequence.
3. Fixed Point Equations
Let be a non-empty set and be a function. Then is a fixed point if .
Now, let be a metric space. is a contraction of if there exists a contraction constant such that:
If is a complete metric space and is a contraction, then there exists a unique fixed point for .