# Math Help - Bounded sequence

1. ## Bounded sequence

So I have this problem I don't know how to solve:

"Show that the following sequences are bounded:"

$U_n = \frac{2n}{3n+1}$

$V_n = sin(\frac{\pi \times n}{2}) + cos(\pi \times n)$

Help would be appreciated!

2. Obviously, $U_n\geq 0$.

We prove that $U_n<\frac{2}{3}$.

$\frac{2n}{3n+1}<\frac{2}{3}\Leftrightarrow 6n<6n+2\Leftrightarrow 0<2$

Then $0\leq U_n<\frac{2}{3}, \ \forall n\in\mathbb{N}$

Sin and Cos are bounded by -1 and 1. Then their sum is bounded by -2 and 2.

Or,

$V_n=\left\{\begin{array}{llll}1, & n=4k\\0, & n=4k+1\\1, & n=4k+2\\-2, & n=4k+3\end{array}\right.$

Then $-2\leq V_n\leq 1, \ \forall n\in\mathbb{N}$