The Eucharist is a reenactment of Christ's last supper on earth before he was betrayed and crucified. If one considers Jesus' words to his disciples at that last supper, then the challenge to reach out to others becomes manifest. It was at that time that Jesus issued a new commandment, that the disciples love one another even as he had loved them. He also said:
By this all people will know that you are my disciples, if you have love for one another.”
Certainly love for one another is the very manifestation of reaching out to others. The entire purpose of the sacrifice of Jesus was to save others, even those who at the time were not believers. The Eucharist therefore challenges one to love one another, even as Christ loved his disciples, and reach out to others even when they may seem unlovable.