Proof the sequence $z_n=px_n+qy_n$ converges implies sequence $x_n$ converges.

Question

I want to prove that the sequence $x_n$ converges, given that the above sequence $z_n=px_n+qy_n$ converges, and that:

• $p,q\ >0$ and $p+q = 1$
• $\displaystyle{ \forall n,\ y_n = \sum_{k=1}^n t_{n,k} x_k }$
• $\forall k,\ t_{n,k} > 0$, $t_{n,k} \xrightarrow[n]{} 0$, and $\displaystyle{ \sum_{k=1}^n t_{n,k} = 1}$