It is human to fall out with each other from time to time, it's natural. We cannot have a relationship with anyone without falling out with them at some point.
The important thing to do, when you have upset someone, or argued, is to be the first person to say sorry. It's simple, yet so few are willing to be the ones to be first to say sorry.
The high achiever, the person who is going places, takes pride in themselves that they don't lose pride or face, if they say sorry first. It doesn't make us feel weak, we can still feel strong, and keep our dignity and our respect.
We say sorry, because we are sorry that we have
got into the argument in the first place. We are really saying sorry
for arguing, because positive, forgiving people know that arguing
doesn't solve problems, or get you anywhere. Despite what the issue
is, we are sorry because we are kind, mature, sensible, and
generous.
Saying sorry has many benefits, even if it is a little difficult at times. Doing so gives you a moral advantage, but also clears the air, gets rid of bad feelings and diffuses the tension associated with the problems.
Remember well, you are not apologising for the issue itself in the first place, but the fact that you had the argument.