2. (a) Let $x, y \in \mathbb{R}$ be such that $x < y$. Prove that there exist natural numbers $m$ and $n$ such that $x + \frac{1}{m} < y - \frac{1}{n}$.
(b) Suppose that $(x_n)$ is a convergent sequence and $(y_n)$ is such that for any $\epsilon > 0$, there exists $K \in \mathbb{N}$ such that $|x_n - y_n| < \epsilon$ for all $n \ge K$. Does it follow that $(y_n)$ is a convergent sequence?
(c) Investigate the convergence of the sequence $(x_n)$ where $x_n = \frac{n^2}{\sqrt{n^6 + 1}} + \frac{n^2}{\sqrt{n^6 + 2}} + \dots + \frac{n^2}{\sqrt{n^6 + n}}$
