This is Hanbun no Tsuki ga Noboru Sora ep 1-6 Complete English Subs
In a winter when he was in the second grade of the high school, Yuichi was hospitalized because of hepatitis. Soon, he became so good as to be scolded by the nurse for going out of the hospital. One day, he happened to meet Rika, a girl of his age, in the next building. He was fascinated by her beauty, but she was very selfish girl. Although she led him by the nose, he began to be attracted by her. However, when Rika told him about her illness, he knew why she acted so selfish; she couldn’t live long.