Compactness, Local Compactness, Completeness

Clearly, every compact metric space is locally compact. I always get confused when completeness is introduced into the picture. Which of the following are true? What are some easy counterexamples to those statements that are false?

1. Every complete metric space is compact.
2. Every complete metric space is locally compact.
3. Every compact metric space is complete.
4. Every locally compact metric space is complete.
5. Every locally compact inner product space is of finite dimension.

1. False: the real line with the usual metric is a counterexample.

2. False: the irrationals are completely metrizable and nowhere locally compact.

3. True: a metric space is compact iff it is complete and totally bounded.

4. False: $(0,1)$ with the usual metric is a counterexample.

5. True; see this question and this question.