A lot of people deal with rejection in different ways. And it also depends on the top of rejection. Were they rejected publicly after proposing? Were they in high school? Were they best friends?
All of those things come into account when assessing how to answer that question. But in a generalization of 'most', is to say that they were upset about the rejection, and still had feelings for the person that rejected them, but they ultimately moved on. Some try to play it off like they weren't really interested in the person who rejected them to maintain their 'cool factor', and others can be really crushed and show it. Some vindictive people might go out and try to ruin the person's life that rejected them by destroying something personal of theirs like a car or vandalizing property, but those are more harsh terms.
I would say that most just take the rejection as a sign that they need to improve themselves, and during that improvement process they find someone else worth pursuing and move on past the person who has rejected them.
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First of all its important to know why you have that fear.
Its basically down to a learned response, meaning its actually programmed into us. When you were younger something or several things (and I don't know exactly what) will have created the fear.
For example, asking a girl to the high school disco, and her turning you down is just one way life teaches you that rejection is emotionally painful.
Now what happens is whenever you are in a situation where you might be rejected your brain starts its internal dialogue, reminding you of the pain you felt before.
The good news is this programming can be undone.
Firstly wear some sort of rubber band around your wrist
Then you need to break your state.
The next time you are in a situation where you might be rejected, and your brain starts giving you all the reasons not to approach, ask yourself "what would I do if I were a super confident person"
This will start to make you think of a positive situation even when faced with a negative.
If you do not manage to approach, pull back the rubber band and snap it against your wrist.
Slowly, your brain will start to associate NOT taking action with pain and will stop giving you the reasons not to, as now it has a pain associated with the inactivity of the situation.
This is not a quick fix, but if applied repeatedly has certainly helped a lot of people to overcome a similar fear.