People dont like backing down or being proven they are wrong, even on internet forums where 90% of us will never meet, we still dont like to admit we are wrong to a total stranger.
There are a number of other very good reasons why people won't change their opinions to match yours.
(1) Your opinion is Crap.
The fact that you can throw up endless arguments (if the earth was round, the Australians would fall off the bottom) does not mean it is right.
(2) Resisting Control
You have almost ordered them to change their opinion to yours.
Now it doesn't matter whose opinion is more likely to be correct.
Most people (sycophants and submissives apart) don't take kindly
to being ordered to change their views. They may interpret it
(and they may be correct in this) that you are trying to establish a
psychological oneupmanship edge to later bully that person about.
"This man may be right, but if I admit that; he will walk all over me."
(3) Several Valid Opinions.
There may be several valid opinions on a topic, each with
its own strengths and weaknesses.
(4) Skill In Debating does not Determine Objective Reality
The fact that from long practice of debating an issue, you can argue the strengths of your own opinion and expose the weakness of the other person's opinion better than they can; doesn't mean that their opinion is wrong; it may mean that you are a better debater (or a lawyer) or more familiar with topic.
(5) Trust
People may believe something because they trust the people who told them that. If they don't trust you; they likely won't believe you, if you argue against them.
(6) Multiple Opinions
Complicated people can often hold several different opinions about the same topic.
(7) Enjoying the Argument
They may deliberately adopt one opinion to provide a debating point.
(8) Artistic Adoption
They may know their opinion is bogus, but stick to it because they
want to be eccentric.
(9) Up Yours Knowall
They may strick to an unlikely opinion merely to counter annoy you.
(10) Logical Consequences
If I was to believe that smoking was bad for me, I would have to stop; and that is difficult because I really crave for a smoke, so best not to believe that smoking is bad for me; then I don't have to change.
Now if I was to admit that slavery was wrong, then I might have to stop raping the black slaves girls on this southern sugar plantation.
(11) Indecision can be Dangerous
Constantly re-evaluating and changing opinion in the light of new arguments can result in paralysis by analysis. Like a car is coming fast to you, do you jump left or right. The important thing is to decide and making any decision may be better than trying to make the best decision.
(12) Negotiation
They detect you want them to change their opinion and are holding
out for a trade, some tangible benefit for aligning their way with yours.