Proves for the sequence $a_n$ with $a_n = \cos\left(\frac{n\pi}{4}\right)$
a)Show that it exist an $m \in \mathbb{N}$ that $a_n=a_{n+m} \forall n \in \mathbb{N}$ and determine $M=\{a_n ; n \in \mathbb{N}\}$
b) Find $\forall a \in M$ a subsequence $\{b_j\}$ of $\{a_n\}$ with $b_j = a \forall j \in \mathbb{N}$.Justify your answer.
c)Does a convergent subsequence $\{b_j\}$ of $\{a_n\}$ with $\{b_j; j \in \mathbb{N}\}=M$ exist?Justify your answer.
a)If $m=0: a_n =a_{n+m}$
If $n=1,m=6: a_n =a_{n+m}$
Do I only have to calculate $ \cos\left(\frac{\pi}{4}\right)= \frac{\sqrt{2}}{2}$ to define M?