Hint - Write down the sequence

-1,1,-1,1,-1,1,-1,1,....

This will go on and on...

So what will you conclude?

To prove it formally, Assume a limit L, prove both a_k and a_k+1 can be <1/2 (or a smaller number) (hence this becomes your epsilon) 'away' from L. Hence a contradiction to the definition of the limit