What a wonderful Mom you are, you gave someone something to live for. My oldest daughter just gave away her little Pom to a neighbor who has epilepsy and she is going through the same issuses as you. I feel that the dog now has a purpose in life, and thats to take care of their new owner. You did the right thing, and in your heart you know it too. Take comfort in knowing that Jake has a new owner who loves him.
