Bits are there, but the bits that come out of the video card may not be the bits that the TV sees. Like another comment says, it's all analog once you're outside of the computer, and the conversion between the two is not perfect.
What digital signaling does is fix some common failure modes of analog. The signal is balanced, so it's more immune to picking up and emitting noise. The electrical signal only has two defined states, so if you degrade the signal in some way (attenuation, noise), you'll probably be able to pick the right state. But, you don't get infinite ability to do that, only more. Eventually, the two states become indistinguishable, and then you can't tell which bit was transmitted to you. (But you can't tell that you can't tell, so you start picking bits at random.)
If we take this to the logical extreme, your argument implies that you can build an infinitely long cable with an infinitely tiny amount of material.
What digital signaling does is fix some common failure modes of analog. The signal is balanced, so it's more immune to picking up and emitting noise. The electrical signal only has two defined states, so if you degrade the signal in some way (attenuation, noise), you'll probably be able to pick the right state. But, you don't get infinite ability to do that, only more. Eventually, the two states become indistinguishable, and then you can't tell which bit was transmitted to you. (But you can't tell that you can't tell, so you start picking bits at random.)
If we take this to the logical extreme, your argument implies that you can build an infinitely long cable with an infinitely tiny amount of material.