Can you prove she moved to Bethany?
Yes, because I can provide proof that Maria Valtorta was a true spokesperson of Jesus, and Jesus showed her scenes from His life on earth, and from them, I learned that Mary of Magdala came to live in Bethany again with her brother Lazarus and sister Martha after her conversion, and that's where she resided when she wasn't traveling with Jesus and the other apostles.
One cannot prove a negative- So I cannot prove Mary Magdala did not live in Bethany.
Actually, proving a negative is not inherently impossible, but it is more challenging than proving a positive, and can be achieved through various logical and empirical methods. But, since you say that you can't prove Mary the Magdalene didn't come to live in Bethany, you can't reasonably say that Mary the Magdalene and Mary of Bethany weren't the same woman.
But she followed Jesus so she lived nowhere.
And assumptions are dangerous to make absolute declarations on.
You say "Assumptions are dangerous to make absolute declarations on" right after making an
assumption that Mary Magdalene lived nowhere after becoming a disciple of Jesus...