# Using Dual Double Fuzzy Semi-Metric to Study the Convergence
## 3 Double Fuzzy Semi-Metric
: _From Example 1, we see that:_  
\[\Psi^{\downarrow}(\mu,x_{p},x_{1};y_{p},y_{1}) \leq\Psi^{\downarrow}(\lambda,x_{2},x_{1};y_{2},y_{1})+\Psi^{ \downarrow}(\lambda,x_{3},x_{2};y_{3},y_{2})+\Psi^{\downarrow}(\lambda,x_{3}, x_{4};y_{3},y_{4})\]3  
_and:_  
\[\Psi^{\downarrow}(\mu,x_{p},x_{1};y_{p},y_{1}) \leq\Psi^{\downarrow}(\lambda,x_{2},x_{1};y_{2},y_{1})+\Psi^{ \downarrow}(\lambda,x_{3},x_{2};y_{3},y_{2})+\Psi^{\downarrow}(\lambda,x_{3}, x_{4};y_{3},y_{4})\]4_From Example 6, we have:_  
\[\Psi^{\downarrow}(\mu,x_{p},x_{1};y_{p},y_{1}) \leq\Psi^{\downarrow}(\lambda,x_{2},x_{1};y_{2},y_{1})+\Psi^{ \downarrow}(\lambda,x_{3},x_{2};y_{3},y_{2})+\Psi^{\downarrow}(\lambda,x_{3}, x_{4};y_{3},y_{4})\]5  
_where:_  
\[\Psi^{\downarrow}(\mu,x_{p},x_{1};y_{p},y_{1}) \leq\Psi^{\downarrow}(\lambda,x_{2},x_{1};y_{2},y_{1})+\Psi^{ \downarrow}(\lambda,x_{3},x_{2};y_{3},y_{2})+\Psi^{\downarrow}(\lambda,x_{3}, x_{4};y_{3},y_{4})\]6  
_It is clear to see that \(x_{n}\overset{M^{\triangleright}}{\longrightarrow}x\) and \(y_{n}\overset{M^{\triangleright}}{\longrightarrow}y\) as \(n\to\infty\) if and only if \(\Psi^{\uparrow}(\lambda,x_{n},x;y_{n},y)\to 0\) as \(n\to\infty\) for all \(\lambda\in(0,1)\). The other convergence presented in Theorem 5 can be similarly verified._  
According to Definition 5, we can similarly define the concepts of four types of the joint Cauchy sequences with respect to \(\Psi^{\uparrow}\). We omit the details. **Theorem 6**. : _Let \((X,M)\) be a fuzzy semi-metric space along with a \(t\)-norm \(*\). We also assume that \(M\) satisfies the rational condition, and that the \(t\)-norm \(*\) is right-continuous at \(0\) with respect to the first or second argument. Let \(\{x_{n}\}_{n=1}^{\infty}\) and \(\{y_{n}\}_{n=1}^{\infty}\) be two sequences in \(X\). Then we have the following properties:_  
1. _Suppose that_ \(M\) _satisfies the_ \(\circ\)_-triangle inequality for_ \(\circ\in\{\bowd,\triangleright,\triangleleft\}\)_. Then the following statements hold true:_ * \(\{x_{n}\}_{n=1}^{\infty}\) _and_ \(\{y_{n}\}_{n=1}^{\infty}\) _are two_ \(>\)_-Cauchy sequences in a metric sense if and only if_ \(\{x_{n}\}_{n=1}^{\infty}\) _and_ \(\{y_{n}\}_{n=1}^{\infty}\) _are the joint_ \((\lambda,>,>)\)_-Cauchy sequences with respect to_ \(\Psi^{\uparrow}\) _for any_ \(\lambda\in(0,1)\)_._ * \(\{x_{n}\}_{n=1}^{\infty}\) _is a_ \(>\)_-Cauchy sequences in a metric sense and_ \(\{y_{n}\}_{n=1}^{\infty}\) _is a_ \(<\)_-Cauchy sequences in a metric sense if and only if_ \(\{x_{n}\}_{n=1}^{\infty}\) _and_ \(\{y_{n}\}_{n=1}^{\infty}\) _are the joint_ \((\lambda,>,<)\)_-Cauchy sequences with respect to_ \(\Psi^{\uparrow}\) _for any_ \(\lambda\in(0,1)\)_._ * \(\{x_{n}\}_{n=1}^{\infty}\) _is a_ \(<\)_-Cauchy sequences in a metric sense and_ \(\{y_{n}\}_{n=1}^{\infty}\) _is a_ \(>\)_-Cauchy sequences in a metric sense if and only if_ \(\{x_{n}\}_{n=1}^{\infty}\) _and_ \(\{y_{n}\}_{n=1}^{\infty}\) _are the joint_ \((\lambda,>,<)\)_-Cauchy sequences with respect to_ \(\Psi^{\uparrow}\) _for any_ \(\lambda\in(0,1)\)_._ * \(\{x_{n}\}_{n=1}^{\infty}\) _and_ \(\{y_{n}\}_{n=1}^{\infty}\) _are two_ \(<\)_-Cauchy sequences in a metric sense if and only if_ \(\{x_{n}\}_{n=1}^{\infty}\) _are the joint_ \((\lambda,<,<)\)_-Cauchy sequences with respect to_ \(\Psi^{\uparrow}\) _for any_ \(\lambda\in(0,1)\)_._ * _Suppose that_ \(M\) _satisfies the_ \(\circ\)_-triangle inequality. Then the following statements hold true:_ * _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_ \(\Psi^{\uparrow}(\lambda,x_{m},x_{n};y_{m},y_{n})<+\infty\) _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.