# Using Dual Double Fuzzy Semi-Metric to Study the Convergence
## 3 Double Fuzzy Semi-Metric
Given any fixed_ \(\lambda\in(0,1)\)_, if_ \(\Psi^{\uparrow}(\lambda,x_{m},x_{n};y_{m},y_{m})<+\infty\) _for all_ \(m>n\)_, then_ \(\{x_{n}\}_{n=1}^{\infty}\) _and_ \(\{y_{n}\}_{n=1}^{\infty}\) _are the joint_ \((\lambda,>,<)\)_-Cauchy sequences with respect to_ \(\Psi^{\uparrow}\) _for any_ \(\lambda\in(0,1)\)_._ * _Let_ \(\{x_{n}\}_{n=1}^{\infty}\) _be a_ \(<\)_-Cauchy sequence in a metric sense and let_ \(\{y_{n}\}_{n=1}^{\infty}\) _be a_ \(>\)_-Cauchy sequence in a metric sense. Given any fixed_ \(\lambda\in(0,1)\)_, if_ \(\Psi^{\uparrow}(\lambda,x_{n},x_{m};y_{m},y_{m})<+\infty\) _for all_ \(m>n\)_, then_ \(\{x_{n}\}_{n=1}^{\infty}\) _and_ \(\{y_{n}\}_{n=1}^{\infty}\) _are the joint_ \((\lambda,<,>)\)_-Cauchy sequences with respect to_ \(\Psi^{\uparrow}\) _for any_ \(\lambda\in(0,1)\)_._ * _Let_ \(\{x_{n}\}_{n=1}^{\infty}\) _and_ \(\{y_{n}\}_{n=1}^{\infty}\) _be two_ \(<\)_-Cauchy sequences in a metric sense. Given any fixed_ \(\lambda\in(0,1)\)_, if_ \(\Psi^{\uparrow}(\lambda,x_{n},x_{m};y_{n},y_{m})<+\infty\) _for all_ \(m>n\)_, then_ \(\{x_{n}\}_{n=1}^{\infty}\) _and_ \(\{y_{n}\}_{n=1}^{\infty}\) _are the joint_ \((\lambda,>,<)\)_-Cauchy sequences with respect to_ \(\Psi^{\uparrow}\) _for any_ \(\lambda\in(0,1)\)_._ * _Suppose that_ \(\{x_{n}\}_{n=1}^{\infty}\) _and_ \(\{y_{n}\}_{n=1}^{\infty}\) _are the joint_ \((\lambda,>,>)\)_-Cauchy sequences with respect to_ \(\Psi^{\uparrow}\) _for any_ \(\lambda\in(0,1)\)_. Then_ \(\{x_{n}\}_{n=1}^{\infty}\) _is a_ \(<\)_-Cauchy sequences in a metric sense and_ \(\{y_{n}\}_{n=1}^{\infty}\) _is a_ \(<\)_-Cauchy sequences in a metric sense. _Suppose that_ \(\left\{x_{n}\right\}_{n=1}^{\infty}\) _and_ \(\left\{y_{n}\right\}_{n=1}^{\infty}\) _are the joint_ \(\left(\lambda,<,>\right)\)_-Cauchy sequences with respect to_ \(\Psi^{\dagger}\) _for any_ \(\lambda\in\left(0,1\right)\) _Then_ \(\left\{x_{n}\right\}_{n=1}^{\infty}\) _is a_ \(<\)_-Cauchy sequences in a metric sense and_ \(\left\{y_{n}\right\}_{n=1}^{\infty}\) _is a_ \(>\)_-Cauchy sequences in a metric sense._ _Suppose that_ \(\left\{x_{n}\right\}_{n=1}^{\infty}\) _and_ \(\left\{y_{n}\right\}_{n=1}^{\infty}\) _are the joint_ \(\left(\lambda,<,<\right)\)_-Cauchy sequences with respect to_ \(\Psi^{\dagger}\) _for any_ \(\lambda\in\left(0,1\right)\)_. Then_ \(\left\{x_{n}\right\}_{n=1}^{\infty}\) _and_ \(\left\{y_{n}\right\}_{n=1}^{\infty}\) _are two_ \(<\)_-Cauchy sequences in a metric sense._  
Proof. : For any fixed \(\lambda\in\left(0,1\right)\), using Lemma 1, it follows that there exists \(\lambda_{0}\in\left(0,1\right)\), satisfying:  
\[\Psi^{\downarrow}(\mu,x_{p},x_{1};y_{p},y_{1}) \leq\Psi^{\downarrow}(\lambda,x_{2},x_{1};y_{2},y_{1})+\Psi^{ \downarrow}(\lambda,x_{3},x_{2};y_{3},y_{2})+\Psi^{\downarrow}(\lambda,x_{3}, x_{4};y_{3},y_{4})\]7  
To prove part (i), we just prove the first case, since the other cases can be similarly obtained. Suppose that \(\left\{x_{n}\right\}_{n=1}^{\infty}\) and \(\left\{y_{n}\right\}_{n=1}^{\infty}\) are \(>\)-Cauchy sequences in a metric sense. Therefore, given any \(t>0\) and \(\delta>0\), there exists \(n_{t,\delta}\in\mathbb{N}\) such that \(m>n\geq n_{t,\delta}\) implies \(M(x_{m},x_{n},t)>1-\delta\) and \(M(y_{m},y_{n},t)>1-\delta\).