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Do you know that $\displaystyle \left| {d(x,y) - d(x_n ,y_n )} \right| \leqslant \left| {d(x,y) - d(x_n ,y)} \right| + \left| {d(x_n ,y) - d(x_n ,y_n )} \right|~~(\#1)$
There is a well-known theorem for metric spaces:
for any points $\displaystyle a,~b~\&, t \in S$ it is the case that $\displaystyle \left| {d(a,t) - d(b,t)} \right| \leqslant d(a,b)$.
Apply that to both terms on the right of $\displaystyle ~(\#1).$