This is a $m + n \geq 200 \implies (m \geq 100)\lor(n \geq 100)$ argument, so my approach was to use contraposition so it is:
$(m < 100)\land(n < 100)\implies m+n<200$
Then, my unexperienced self with proofs (yet) decided to use some values and substitute for the greatest possible values for $n$ and $m$ which are supposedly $199$ if the domain is integers or $199.9999999$ if it isn't (I really was experimenting with the question as it had not assumed a domain) and said that since the sums of the two biggest values of $n$ and $m$ is in fact less than 200, then my contrapositive argument is true and therefore the main argument is also true. My proof was written in a sloppy manner because I didn't know how to formulate it but this is exactly how I solved it.
Is this correct? If it isn't, then how should it have been done? (or even if it is, I am sure there is a more systematic approach to it, I'd be happy to know what it is)