Sympathy is the act of sharing the feelings of one who is experiencing sadness. Encouragement is the act of supporting a person, typically when they are feeling down, and helping them to move forward through one's words and actions. The person who expresses sympathy and the person who gives encouragement to someone in actuality have the same purpose. Both acts attempt to authenticate a relationship between the giver and the receiver. The receivers should thus be assured that someone is there for them during this tough time. Below, I have listed some examples of how giving sympathy is just like giving encouragement:
Example 1:
If a child loses his first soccer game, he is likely to be upset. If he had practiced after school everyday and woken up early in the morning to practice in the backyard, the loss might be more devastating to him. As a parent, or even simply as a friend, you can sympathize with him and share the feeling of sadness. You seem to be able to feel how he feels even if you are not the one experiencing it. It does not stop there, however. After expressing that you understand how much he wanted to win the game and how hard he practiced to obtain that victory, you can offer encouragement. You can say that he can use this loss as motivation to strive harder and win the next. Certain strategies can be used which were not initially implemented.
Example 2:
A rejection from the college of one's dreams, is extremely bad news for any high school student. If they studied hard to achieve good grades and involved themselves in extracurricular activities, not getting in may leave them confused and discouraged. One may sympathize with them, and picture themselves not having gotten into their dream school, and the same emotions might be provoked. The rejected student's story may affect the listener, but the listener must be strong and provide words of encouragement. One can say that if they really want the school to be the one they end up at, they should not give up. One can suggest that they can appeal or even decide to go there after two years as a transfer student. Maybe even opening their eyes to the schools they did get into will help them to realize that those schools are just as good.
With both these examples, sympathy and encouragement come hand in hand. While you could just sympathize with the person, making the effort to actively encourage them and helping them to realize that there will be better ends than what they think the results will be, is a much better route. Positivity is key. Because it may be difficult to find this in a time of sorrow, you as a friend could help lead them to it.