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I’ve done this with Delta a few times but rarely does the math work.
Usually, at least with Delta, the First Class upgrade price is ridiculous compared to just booking First Class in the first place (sometimes the upgrade is more than just buying a First Class ticket… why? Delta).
Once in a great moon though I can book a points ticket in Main Cabin or Comfort Plus and then I keep checking the First Class cash upgrade and will find a sweet deal. But let me also say those deals don’t last.
The last time I did this on an MSP to SEA flight, I was able to snag a cheap upgrade but after I bought mine it drove the algorithm to push the price 300% higher.