Sentence I: This implies a small amount of hope, but in common usage, it often conveys a pessimistic tone, suggesting almost no hope. This is the correct usage. 

Sentence II: The addition of the article "a" makes this more positive, indicating a small but definite amount of hope, matching the meaning "some hope at least." This is thus also correct. 

Sentence III: The phrase with the article "a" typically refers to someone living near another person, which fits the meaning "living near one." This is also correct. 

Sentence IV: 

Since all the statements are grammatically correct, the correct option is D.