- #1

MathematicalPhysicist

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1)(1+1/n)^n<3 for every n>=3.

2) (x^n+y^n)/2>=((x+y)/2)^n for every n natural and every x,y>=0.

3) |a+1/a|>=2 for every a different than 0.

for the first i thought to use induction and to use the fact of increasing sequence (1+1/n)^n or of the decreasing sequence (1+1/n)^n+1, but i didnt get much far, i also tried to use the fact of 1/n<=1/3, but im stuck.

for the second i tried by induction: ((x+y)/2)^n+1<=(x^n+y^n)/2(x+y)/2 but im also stuck.

for the third i tried dissecting it into parts, i.e 0<a<1 a>1 or -1<a<0 a<-1 but i got stuck in a>1.

i mean for a>1 we can write a=h+1 when h>1 and thus h+1+1/(h+1)>2+1/(h+1)>2 but im having a problem with 0<h<1 then 1<h+1<2 1/2<1/(h+1)<1

3/2+h<1+h+1/(h+1) but form here i couldnt conclude that it's bigger than 2.