Most of the time when someone writes a final, one-sentence "Period" or "Full stop" (with or without drama and a weighty silence) they should take it out. "Period" is not much of a statement. I don't mean logically, only, but interpersonally. "Period" means you're gloating over your rightness. It calls to mind what the villain does at the end of the movie: they would have won, but they have to stop and gloat. So they lose. Period.

(It works better in speech, but only about half the time.)

This isn't some sort of grammar rule - nothing like it. But you rarely prove much by ending on a note that says "I'm self-satisfied and closed-minded." It probably will (and probably should) sound complacent rather than thorough and convincing. The irony is that speakers usually say "Period" about where they usually say "ecksetera" - when they've run out of ready examples and want to sound as if they're just getting started. If you give your reasoning in two highly debatable or unsubstantiated claims followed by "Period," that's par for the course, or sadly even better than average.

It also sounds unintentionally comical. You... um, didn't know you aren't supposed to read that part out loud? Oh! You're done talking. Oh. Ok. That's very interesting that you're done talking. Stunning announcement, really. Oh, wait... you mean it proves you were right? Good Lord, I must be in the wrong building. What's that? This is a mental hospital, you say, not the middle horse stables? Oh yes, quite, quite. Beg pardon. Will be on my way now. No trouble. I'll get the door myself.

It's a style thing. I'm just pointing out why I think it's poor style.

All I'm saying is: remove it and try that. If it needs something, try something else. Try not to punt by spelling out a common punctuation mark. That's like yelling when someone challenges your view with a fact: more volume isn't relevant unless we're having audio trouble, and it probably seems foolish or offensive. I know what a period is and how to spell the word. Also, I heard what you were saying before that. If you really want to call attention to a logical argument that is now, you believe, complete, and bracket it verbally, "QED" is a little more on point. Or if you want to be more colloquial: "I rest my case."

But "Period" tends to appear where "QED" or "I rest my case" wouldn't be appropriate, because the argument is not as resounding or airtight as the speaker would like you to believe. "I rest my case" projects the confidence of readiness for any objection that might come up. "Period" gives the impression of trying to silence any objection without having earned that respect.