If (you know they're lying) they've already been caught!
Unless you have a history of being paranoid or insecure there's no reason for you not to listen to your gut instincts. Never ignore "red flags". (Assume you are correct)
If something doesn't (feel right to you) it's probably not right for you!
With regard to "catching' someone lying or cheating or whatever that's pretty much an ego game where someone wants to show the other person how "smart" they were. They're hoping to gain some satisfaction in seeing them react, squirm, or be shocked.
However if the offence committed is a "deal breaker" it makes no sense to invest another minute of your time being around her or him. In order to let go you have to move on.
Truth be told most liars will trip themselves up without any assistance.
The real question is:
"Why do (you need) to catch a liar red handed?
There are only two outcomes either you intend to end your association with them or you plan to continue it (hoping they are going to "change.)
Neither result will bring you joy in the end.
People don't "change" unless (they) are unhappy.