The original statement is equivalent to "It is not the case that there exists 0 such that for all x, x+0=x", so, to form negation, you just remove "It is not the case that".There does not exist 0 such that for all x, x+0=x
Negation: For no 0 does there exist x, where x+0 =/ x
Should be "accurate or entertaining".The newspaper article was neither accurate nor entertaining.
Negation: The newspaper article was accurate and entertaining.
You are making this more complicated than necessary. "There exists epsilon > 0 for each N" means "For each N there exists an epsilon > 0 (which may depend on N)", and it's a wrong way to start the negation. Rather, you keep the same order of variables: epslion, N, n, but change the quantifiers into the dual ones. "There exists an epsilon > 0 such that for each N in the Natural Numbers, there exists an n >= N such that |An - L| >= epsilon".For each epsilon > 0 there exists N in the Natural Numbers such that for all n >= N, the abs|An - L| < epsilon
Negation: There exists epsilon > 0 for each N in the Natural Numbers such that there exists n >= N, |An - L| >= epsilon
The rest seems good to me.