It's true that many people straight and gay alike will try to convince a bi person that they are either gay or straight. The truth is that no one knows whether you are or not, other than yourself, and why try to prove it? In my experience, people generally stick to what they believe, and it's given me peace to let go and allow them to believe what they want, all the while knowing what my own truth is.
People tend to have a preoccupation with your "present" situation. If you're with the same sex, you're homosexual; with the opposite sex, heterosexual. This is problematic since this leads to bi erasure and leaves you feeling like you have to prove that you like both, even if you haven't dated at all or only dated one gender. Your sexuality is valid.
You don't have to. Maybe because they don't believe you, but that's inappropriate. Whoever you tell should believe you. But you can always just say, "No. I'm Bi, and I don't need to prove it to you because I'm telling you now that I am."
There has been a lot of misunderstanding between being gay/lesbian and being bi that there are times when you tell people that they are Bi they would normally ask you what it is and how does it work. Just feel free to feed the curiosity of other people and don't be enraged when they don't completely get the idea and understand that they do not know what it's like to be one.
The answer is....you don't! It's either believe it with your own eyes or not! A good example of this is with those who believe in God. They believe in something they never saw before by writing or conversations, or there are others that don't believe because it isn't "proven enough" to them. You can feel free stating you are Bi, but proving it every time is just annoying for you I see, and therefore you are just coming across ignorant fellows. :)
Being Bisexual, even in the LGBTQ+ community, is often seen as "not picking a side" or "being confused" and often people do not believe it because they do not understand the fluidity in their own sexuality. This being said you should never feel like you have to prove anything to anyone regarding your sexuality or orientation.
People who only experience attraction to one gender can find it a little hard to understand that some people like more than one gender. There's also the misconception that bisexuality is a phase due to people experimenting with their sexuality usually adopting the bisexual label. Sometimes the label fits, sometimes they end up going back to the straight label or going to the gay label. For those who don't understand how someone could like more than one gender, they may believe it's made up because it doesn't fit their frame of logic. There are a lot of misconceptions around being bisexual, so sadly many have to constantly prove themselves about it.