Given a sequence $b_n = \sin(\frac{n \pi}{2})$, I am trying to show that $(b_n)$ diverges.
I have the idea down, I know exactly what to do, but just not HOW to do it.
I know that any convergent sequence, has all sub-sequences converging to the same limit.
So all I need to do is show two sub-sequences which converge to different points.
This is easy, but I am not sure how to express it.
I want to use the notation that a sub-sequence of $(b_n)_{n \in \mathbb{N}}$ is $(b_{n_k})_{k \in \mathbb{N}}$
how can I express the sub-sequences of $\{1, 1, 1, \dots\}$ and $\{0, 0, 0, \dots\}$ ?
The former occurs for every 4th value of $n$ beginning from $n=1$ and the latter from every 2nd value of $n$ beginning from $n=2$.
Could I write the former as the sub-sequence $(b_{4n+1})$ and the latter as $(b_{2n})$? would this be correct notation?
Hint:
an even integer is of the form $n=2k$, $k \in \mathbb{N}$, an odd integer has the form $n=2k+1$; for $k$ odd or even your $sin$ function change sign, so you have also to separate the two case for $k$ (in the same manner) .