I would give Vitamin D as well. Unless you live in a sunny, warm area where you can sunshine all winter.Â
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Last winter was the first winter we started supplementing with D, and we did experience a decrease in the amt. of colds and respiratory infections. I did supplement "conservatively"...I think I could have given more,.
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This past year we moved from New England to southern Fl.  It is now Nov 24th and my kids (and I) haven't had a single cold since coming down here. When we lived up North, we always had at least one cold in Oct or Nov,.  I attribute this to the fact that we are now in a sunny, warm spot and spend lots of time outside w/o sunscreen so we get tons of Vitamin D exposure.    I think the amt. of Vitamin D made from sun is so much more than the amt. people get from food or supplements.  Like the body makes something like 8000 IU of Vitamin D with 20 minutes of sun exposure at mid-day.
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The more i learn about Vitamin D, the more i learn how deficient most people are. Our modern lifestyle with spending so much time indoors is just so different to the way we evolved.