Let

S={xinR|x^(2)

.\ (a) Show that if

M>0

and

M^(2)>=2

, then

M

is an upper bound for

S

.\ (b) Show that if

1

then

M

is not an upper bound for

S

.\ Hint: let

\\\\epsi =2-M^(2)

and

\\\\delta =(\\\\epsi )/(2M+1)

. Show that

\\\\delta

and then consider

(M+\\\\delta )^(2)

.\ (c) Prove that there exists

ainR

such that

a^(2)=2

.

Let S={xRx22}. (a) Show that if M>0 and M22, then M is an upper bound for S. (b) Show that if 1M2