As a consequence of Intermediate Value Theorem (IVT), it is illustrated in almost all calculus books and notes/courses that every polynomial of odd degree over real numbers have a real root.
I tried to see whether this result has been proved without the use of intermediate value property, and I did not find any link or question related to that on this and other sites.
Can one suggest me whether there is a proof of above mentioned consequence of IVT, which is done without using IVT?
I had seen that the fundamental theorem of algebra have been of interest to many people for proving just using algebraic techniques, but among all the known various proofs of FTA, epsilon amount of analysis is used.
So, my above question is of similar nature, but I did not see any comments on its proofs without IVT.