My sister's Swedish Visa was refused. My elder brother is a Swedish citizen and wanted to sponsor her, but the embassy said there is no proof that they are siblings, despite the fact that they bear the same surname. My question is how can she prove they are siblings?
You would typically prove that with some sort of birth certificate or a similar document stating that you have the same parent(s), i.e. that your father and/or your mother have the same name, date and place of birth. Obviously, you would have to provide a birth certificate for the sponsor and another one for the applicant. In some countries, it's also possible to submit a copy of a family register that would document your relationship even more directly.