Prove or disprove thereexists x elementof R forall y e R y2

Prove or disprove: thereexists x elementof R, forall y e R, y^2 > 2013 + x Prove or disprove: thereexists x elementof R, forall y elementof R, y^3 > 2016 + x Prove or disprove: forall elementof > 0, thereexists N elementof N elementof Z^>0, n greaterthanorequalto doublesidearrow 1000/n

Solution

a)

True

We know y^2>=0 for all y in R

SO, choose, x=-2017

So, 2016+x=-1

b)

False.

f(x)=x^3 has range from -infinity and infintiy ie unbounded from above and below

So no such x exists

(c)

e>0

So, there exist N so that

1000/e<N

Hence for all n>=N

1000/e<n

Hence, 1000<ne

1000/n<e for all n>=N

Hence proved