Show that the set of real numbers contains element x, such that x² = 2?

Question

Show that the set of real numbers contains element x, such that x² = 2?

Answer 1

yes it is!!

Answer 2

Theorem:  Every positive number has a square root.  In other words, if α > 0, then there is some number x such that x² = α.

Proof:  Consider the the function f(x) = x², which is certainly continuous.  Notice that the statement of the theorem can be expressed in terms of f:  "the number α has a square root" means that f takes on the value α.  

There is obviously a number b > 0 such that f(b) > α;  in fact, if α > 1 we can take b = α, while if α < 1 we can take b = 1.  Since f(0) < α < f(b), applying the Intermediate Value Theorem to [0, b] implies that for some x (in[0, b]), we have f(x) = α, i.e., x² = α.  Ã¢Â– 

Answer 3

real numbers are from -infinity to +infinity

so for x^2=2

         x=+sqrt(2) and x=-sqrt(2)

so real number also contains these two numbers