# Show that there exist unbounded sequences, $x_n\neq y_n$, such that $x_n-y_n\rightarrow 0$ as $n\rightarrow \infty$

Show that there exist unbounded sequences, $x_n\neq y_n$, such that $x_n-y_n\rightarrow 0$ as $n\rightarrow \infty$

How do I prove this? What sort of sequence would even satisfy this, if it is unbounded?

Work backwards. Choose a sequence $z_n$ that tends to 0 and any unbounded sequence $y_n$. Define $x_n = y_n + z_n$. Then by construction $x_n - y_n = (y_n + z_n) - (y_n) = z_n \to 0$.
The simplest is perhaps: $$x_n=\sqrt{n+1}, \quad y_n=\sqrt n.$$
• what about $n + \frac1n$ and $n$? :) – Giovanni Oct 3 '15 at 19:21
take $x_n = n + \frac{1}{n}$ and $y_n = n$, then $x_n - y_n = \frac{1}{n}$