As Antons says, one has to differentiate between the abstract mathematical definition of infinity and physics.
Can you say that the number zero has a physical existence?
Your reasoning was already the point Planck made about light being a wave carrying energy. There is an infinite number of wave lengths and therefore, the electro-magnetic energy of say, a star, would empty itself instantly if light was a wave. Hence the idea of the quantum physic: Light is small cells of energy with higher energy at higher frequency. Today, we reckon that light can be considered both as a wave and as photons of given amount of energy.