Counterexamples about locally compact sets on the real line

Question

Is there a counterexample in the space $\mathbb{R}$ with it's usual metric to the statements:

• The union of two locally compact subsets of $\mathbb{R}$ is locally compact
• The complement of a locally compact subset of $\mathbb{R}$ is locally compact

My textbook says both statement are false, but I cannot think of a counterexample for neither.

Answer 1

1. $A = \bigcup_{i=1}^{\infty} \left(\frac{1}{2i+1}, \frac{1}{2i}\right]$, $B = [-1, 0]$

2. $A$ in answer 1.

   I think these would work.

*Edited : $A$ was originally an open set, whose complement in $\mathbb{R}$ would be closed and thus locally compact. Thanks to clark's critique, the example was modified.