Why is the limit of a sequence wrong based on the following condition? Let $\{b_{n}\}^{\infty}_{n=1}$ be $$b_1=1$$
$$b_{n+1}=1-b_n$$ 
Here $n\in \Bbb{N}$. We want to calculate the limit of the sequence, call it $M$. Then $$\lim\limits_{n\to \infty} b_{n+1}=\lim\limits_{n\to \infty} 1-b_n=\lim\limits_{n\to \infty} 1 -\lim\limits_{n\to \infty} b_n$$. 
Then because $\lim\limits_{n\to \infty} b_{n+1}=\lim\limits_{n\to \infty} b_n= M$, then $M=1-M$, then $M=\frac12$
Apparently the result is wrong because $\{b_{n}\}^{\infty}_{n=1}$ isn't necessarily convergent so $\lim\limits_{n\to \infty} b_{n+1}=\lim\limits_{n\to \infty} b_n= M$ is likely wrong. But I do not have a compelling reasoning. Could someone give a clear one?

Actually, you will see that this sequence is "trivial" after some computations. $b_1=1, b_2=0, b_3 = 1, b_4 =0, etc ...$ Your sequence is thus $$1, 0, 1, 0, 1, 0...$$ and so on. Obviously it does not converge.
