The question reads:
"Suppose that the largest number of points that any horizontal line intersects the graph of a polynomial f(x) is m. Prove that the degree of f(x) is at least m".
I understand the proof graphically - the degree of a polynomial determines the number of local maximum or minimum points minus (-) one (1). If the degree is 4, there are 3 local maximum/minimum points when the function is graphed. And the curves of the graph that make those points allow for a horizontal line to intersect it 4 times at some specific y value, 4 being also the degree of the function. But how does one prove what is asked in the question algebraically? Please help. Thank you very much if you decide to!