Think about it this way: A person of good, upstanding character won't cheat, whether they're bisexual, homosexual, or heterosexual. Sure, some people may transition that way, slowly realizing or letting themselves realize they're gay—but others are aware of themselves and know that they're bisexual without question. When it comes to marriage, a bisexual person may end up with a partner of either gender. Don't let paranoia ruin the relationship's potential. If you don't feel reassured and safe, it may be a relationship that needs to end.
Know that many bisexual people will not prefer one sex over another.
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You wouldn't discriminate against someone of a different race or denomination, so how is this any different? That is, it doesn't really matter. They are just like everyone else. Not ignore it or be threatened by it. Know that they're just as likely to be monogamous. If at some point in the future you realize that you were wrong, that's completely fine -- there's nothing wrong with that, and nothing wrong with you.