a^2+b^2+c^2=1,(a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca因为2ab≤a^2+b^2,2bc≤b^2+c^2,2ca≤c^2+ a^2,所以2ab+2bc+2ca≤2(a^2+b^2+c^2),从而(a+b+c)^2≤3(a^2+b^2+c^2)=3.(a=b=c时取到等号)