7．
如图，已知曲线\({C}_{1}:y= \dfrac{2x}{x+1}(x > 0) \)及曲线\({C}_{2}:y= \dfrac{1}{3x}(x > 0) \)，\({C}_{1} \)上的点\({P}_{1} \)的横坐标为\({a}_{1}(0 < {a}_{1} < \dfrac{1}{2}) .\)从\({C}_{1} \)上的点\({P}_{n}(n∈{N}^{*}) \)作直线平行于\(x\)轴，交曲线\({C}_{2} \)于\({Q}_{n} \)点，再从\({C}_{2} \)上的点\({Q}_{n}(n∈{N}^{*}) \)作直线平行于\(y\)轴，交曲线\({C}_{1} \)于\({P}_{n+1} \)点，点\({P}_{n}(n=1,2,3⋯) \)的横坐标构成数列\(\left\{{a}_{n}\right\} \)．
\((1)\)求曲线\({C}_{1} \)和曲线\({C}_{2} \)的交点坐标；
\((2)\)试求\({a}_{n+1} \)与\({a}_{n} \)之间的关系；
\((3)\)证明：\({a}_{2n-1} < \dfrac{1}{2} < {a}_{2n} \)．