Let $(x_n)$ be a real valued sequence. I want to construct a subsequence $(y_n) $ of $(x_n) $ such that no two consecutive terms of the sequence $( y_n) $ are same and also if $(y_n) $ is Cauchy then $(x_n) $ is also Cauchy.
Intuitively it is very clear that whenever there are two consecutive equal terms in $(x_n) $ say $x_m=x_{m+1}$, then I won't consider $x_{m+1}$ in $(y_n) $. How should i write this sequence $(y_n) $ mathematically?
As has been pointed out, that isn't always possible. The case where it isn't possible is when $(x_n)$ is an eventually constant sequence. A sequence $(a_n)$ is eventually constant if there exists an integer $N$ and a value $c$ such that $\forall n \ge N: a_n =c$.
It's obvious that you can't find an (infinite) subsequence $(y_n)$ that isn't also eventually constant, contradicting your condition $\forall n:y_{n+1} \ne y_n$.
If $(x_n)$ isn't eventually constant, then one possible index sequence $n_k$ for the subsequence $(y_k) = (x_{n_k})$ can be defined inductively step by step as:
$$n_1=1, \forall k\ge 1: n_{k+1}=\min \{i\; |\; i>n_k, x_i \neq x_{n_k}\}$$
$(x_n)$ not being eventually constant makes the definition of $n_{k+1}$ (the smallest index following $n_k$ where the $(x_n)$ sequence takes a different value than $x_{n_k}$) valid, there always exists such an index.