So I have this problem I don't know how to solve:
"Show that the following sequences are bounded:"
$\displaystyle U_n = \frac{2n}{3n+1}$
$\displaystyle V_n = sin(\frac{\pi \times n}{2}) + cos(\pi \times n)$
Help would be appreciated!
Obviously, $\displaystyle U_n\geq 0$.
We prove that $\displaystyle U_n<\frac{2}{3}$.
$\displaystyle \frac{2n}{3n+1}<\frac{2}{3}\Leftrightarrow 6n<6n+2\Leftrightarrow 0<2$
Then $\displaystyle 0\leq U_n<\frac{2}{3}, \ \forall n\in\mathbb{N}$
Sin and Cos are bounded by -1 and 1. Then their sum is bounded by -2 and 2.
Or,
$\displaystyle V_n=\left\{\begin{array}{llll}1, & n=4k\\0, & n=4k+1\\1, & n=4k+2\\-2, & n=4k+3\end{array}\right.$
Then $\displaystyle -2\leq V_n\leq 1, \ \forall n\in\mathbb{N}$