Normally you'd be right, urushibara. "ene" does normally imply double bond. In this case, however, ethylene is the monomer. The pi-bond in one ethylene molecule breaks and connects to the next one, forming a sigma-bond between them, which causes the pi-bond in the second to break, which joins to... and so on. The equation would be written:
x(H2C=CH2) --> ...-CH2-CH2-CH2-CH2-...
The name of a polymer like this tends to be written Poly-<insert name of monomer here>. For example, Poly Vinyl Chloride:
x(H2C=CHCl) --> ...-CH2-CHCl-CH2-CHCl-...
Or even Poly Tetrafluoroethylene (Teflon):
x(CF2-CF2) --> ...-CF2-CF2-CF2-CF2-...
Does that make it clear? I'd post a reaction mechanism (complete with curved arrows!) to make it clearer, but you can find them in an average text book, plus I have no idea of how to post pictures!