That is unfortunate. What is your question?

i guess it should be , why is it happening?

they seemingly just cheat on me , and i have wasted lots of my time for that

they could have just been straight up rude in rejecting me , but they just gone up and lie at me

Ok, so the way I see it is this. You contacted a faculty and were pushing them for support. They likely saw you as not acceptable for the program, but because you may have insisted, they likely sarcastically said if you can solve this problem that is so beyond your scope and capability, you can get admitted. They basically rejected you, but put it on you, not on them. You were rejected for failing when you were set up to fail. Perhaps they thought they were being funny, or that you would take a hint and move on, perhaps you pressed them too hard for a position and they were annoyed.. who knows

cont... how do you avoid this next time? When you contacted them, did you press for a position or enquire politely (no big praise or superlatives, just a professional cordial email?) Was your application package suitable for the ranking of the University? i.e. did you stand a reasonable chance or was it a longshot to begin with? When you were assigned this task, did you seek out some council from your undergraduate professors to see if it is reasonable?

i think that uhm i didn't push them , i just keep updated when i reviewed my proof , and i didn't really push them , but merely questioned about the faculty , and they are like acting to be cool and willing to accept up to the point where i presented a perfect proof , and they just like showing their true color , but never mind , i will just move on

but uh ... idk coz i don't think that the email could provide emotional contact at all ? coz by words , how could one just access for others feelings

but never mind , i am not angry with them at all , love and peace anyways

i just applied for phd , instead of some position though

is this question so hard at all ?

There is a lot to unpack in this question and in the comments on @buffy's answer. In particular, you don't give any details on the interaction with the professor who you claim "lied" to you and "cheated". As a consequence, we cannot say much about what that person did or didn't do, and whether that was ethical.

But I can say something about the documented you posted at researchgate.net/publication/… . You get straight to the point that the claim Cantor made some 150 years ago about the cardinality of the integers is wrong. I do not need to check the details: If the best mathematicians, over the course of these 150 years, have accepted Cantor's argument as true, a two-paragraph statement to the contrary cannot be true.

The short answer then is that your paper is "quack science", and I would suggest you take a look at this question on this forum, along with the many highly rated comments and answers: academia.stackexchange.com/questions/18491/… My suggestion therefore would be to critically consider whether the problem could, just perhaps, be with you instead of the professor you've been interacting with.

yo ... claiming that it is pseudoscience before i have rigorously proved it ? it can't be right but you made to the point , it might be that if all mathematics are wrong then their work would be in trouble , but i think it is more of truthfullness or poltical correctness thingy