Does every convergent sequence have a sub-sequence whose terms comes closer than any positive sequence? Let $(x_n)$ be convergent sequence of real numbers and $(y_n)$ be any sequence of positive real numbers , then is it true that there is a sub-sequence $(x_{r_n})$ such that $|x_{r_{n+1}}-x_{r_n}|<y_n , \forall n \in \mathbb N$ ?

By definition of convergence (actually, by using the fact that any convergent sequence is a cauchy sequence), $\forall \epsilon > 0, \exists N\in \mathbb{N}: \forall n, m > N, |x_n - x_m| < \epsilon$. So, if you have a sequence $(y_n)$, take the $y_1$ as epsilon, choose such $N_1$ that the statement above works: $$\epsilon = y_1, \exists N_1\in\mathbb{N}: \forall n, m > N_1, |x_n - x_m| < y_1$$ Now, choose $r_1 = N_1+1$. Continue with $y_2$ the same way, just remembering that you can't choose $r_2 < r_1$: $r_2 = N_2+1$, where $$\exists N_2\in\mathbb{N}: \forall n, m > N_2, N_2 \geq N_1, |x_n - x_m| < y_2$$ You'll get your subsequence $(x_{r_n})$ in question.