i think, this has been solved before.. anyways,

and you want

.. the same way for x approaching neg infty.

now, if n is odd, limit as x approaches positive infinity is 1 while the limit as x approaches negative infinity is -1 (show it) according to a corollary to IVT, if a fuction is continuous on [a,b] and , then there is a c on (a,b) st. f(c) = 0..

using this, let a be a very "large" negative number and b be a very large positive number.. then , since f(a) is "almost" -1 and f(b) is "almost" 1. hence there is a c in (a,b) s.t f(c) = 0; and this c is a real root (zero).. QED