Proving a Sequence Does Not Converge I have a sequence as such:

$$\left( \frac{1+(-1)^k}{2}\right)_{k \in \mathbb{N}}$$

Obviously it doesn't converge, because it alternates between $0,1$ for all $k$. But how do I prove this fact?
More generally, how do I prove that a sequence does not converge? Are there any neat ways other than "suppose, for contradiction, that the sequence converges. Then..."?

$\{a_k\}$ converges if and only if $\{a_{2k}\}$ and $\{a_{2k+1}\}$ converge to the same limit. Here
$$
a_{2k}=1\rightarrow1, \quad a_{2k+1}=0\rightarrow 0.
$$
