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>Today for the duration of ~40 mins I ran until I was tired, and walked till I could run again. Is that a good approach?
yes. keep doing that until you can run for the entire time you want to run.
you should stretch after running.
>Should I breathe through nose or mouth also when I'm tired?
you should breathe through your nose as much as possible, but if you feel you aren't getting enough oxygen you can breathe through your mouth.
i think you should do more than just pushups on the days you don't run. but you don't need to go crazy.
you really need to watch your diet if you want to lose or maintain weight, exercise is not enough.