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?

What you proved is this:

If the sequence $b_n$ is convergent, then its limit must be equal to $\frac12$.

What you did not prove is:

The sequence is convegent

Meaning the statement above cannot help you. This, in itself, does not mean that the sequence is not convergent. The proof is wrong, yes, but having an incorrect proof of a statement does not mean the statement is wrong. 

For example, I can say "All cats have a brain, I am a cat, therefore I have a brain". This is clearly wrong, since I am not a cat (or am I), but we cannot conclude from that that I do not have a brain. The statement "I have a brain" is still unproven, but that in itself doesn't mean it's wrong.

In your case, however, the statement is wrong. $b_n$ is not a convergent sequence. You still have to prove that, and it's easy if you write down a couple of elements in the sequence.
In fact, the sequence you have is the sequence $$1,0,1,0,1,0,1,0,1,0\dots$$
which is clearly not a convergent sequence.
