It won't kill it outright, but cutting too short does weaken a lawn. Best thing at the end of the year is to let it grow a little longer than usual, and leave it alone. Maybe rake it to pull out the dead thatch.
A light fertilizer app can help if your soil isn't too good, or you remove clippings, which steals away the nutrients.
If you cut too short you will get a lot more weeds and weedy type grasses growing. Most people cut too short all the time. Cut it on your normal schedule, but with the mower raised a notch or two higher.
You will notice it stays greener and doesn't burn out in the summer as much.
You should really cut only the top third of the grass on a regular basis in order to keep your lawn healthy. Much lower than this and you expose it to shock and the possibility of disease, fungus and weeds/moss getting into the lawn.
If you already have these issues, raise your cutting height until you are just clipping the top of the grass and mow regularly to promote new growth (and water well). If you have weeds/moss, scarify and perhaps aerate the lawn before feeding it to ensure it has the right nutrients to bounce back!
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