Hundred of frustrated people are looking for ways to "win my ex back". Breakups are often short term solutions to long term problems and it is not uncommon to want to get back together again.
Relationships can go on for a long time even when they are not working very well. People are just too busy to either break up or fix the relationship.
The fact that you are apart gives you the chance to sit down and really analyze what the problems were.
The final argument is often not the main problem, more than likely it was only the final straw that forced the decision.
Once you have thought about the real causes of the breakup you have to then resolve them. If it was something that could be solved with an apology, then you should write a letter of apology. Take your time and make sure that what you write is genuine and your ex will appreciate the effort you have put in.
If you tell them that you can understand why breaking up was the best thing to do, their reaction will surprise you. They will not be expecting this approach; they will have been expecting an argument and would have been ready to be on the defensive. Instead they will be met with an agreement and an acknowledgement that there was a problem. Then you can start talking about finding solutions.
Your ex needs to know how important they are to you, this is often the beginning of all the problems in the first place, they have felt neglected and alone.
Any other issues will also have to be dealt with before you can get back together.
Getting your ex to start to trust you again takes time. Start by taking small steps, perhaps just meeting for a chat, it is important that they don't feel you are putting pressure on them.
Learning to make your relationship your Number 1 priority and convincing your ex that you intend to keep it that way will pave the way for a new beginning.
