How do you prove you're own existence

Basically, you can't.

It's a philosophical question, and it can be rephrased as : how can you know that others exist? (that you're not dreaming them for example) : you can't. And so, you can't really prove to others that you exist.

Different people tried differently, for example Descartes said that he knows he exists because he thinks.

In practical terms, your actions prove your existence.