I'll try to answer the question directly, without simplifying to a smaller case.
Proof with recursion, rather than induction
If there are 100 blue eyed people on the island, person 1 sees 99 blue-eyed people.
Let's number them person 1 through person 100 (without loss of generality).
Person 1 imagines that person 2 sees either 99 blue eyed people (case 1A) or 98 blue eyed people (case 1B).
Person 1 imagines that in the case 1B, person 2 imagines person 3 sees either 98 blue eyed people (case 2A) or 97 blue eyed people (case 2B).
Continue this chain of recursion 99 times and you get:
Person 1 imagines that person 2 imagines that person 3 imagines... ... ...that person 97 imagines person 98 imagines that person 99 imagines that person 100 sees either one blue eyed person (case 99A) or zero blue eyed people (case 99B).
After night one, person 1 (which, since this was done without loss of generality, is all 100 people) can all simultaneously agree that everyone else is no longer imagining a hypothetical where there is only one person.
Because of this, they all now know that case 99B is impossible - more importantly, though, they know that everyone else knows case 99B is impossible.
Why does this work?
Each person does not know the color of their own eyes.
When these people imagine another person's perspective, they must imagine two hypotheticals - one where that person sees the imaginer's eyes as blue, and one where that person sees the imaginer's eyes as non-blue. Recursively, when they imagine another person imagining yet another person's perspective, they must then imagine two perspectives again, for a total of four possibilities. This would grow exponentially, but the solution of the puzzle only requires them to expand one arm of the tree.
This means that when person 1 imagines person 2's perspective, there are two hypotheticals: either person 1's eyes are blue, or they are not blue. There are two people's eyes who are indeterminate in this hypothetical. Person 1 still doesn't know what color their eyes are, so they must imagine multiple hypotheticals for what person 2 might be deducing from.
Similarly, when person 1 tries to imagine what person 2 might imagine person 3 thinks, person 1 must now evaluate four cases again - either person 1's eyes are blue or not blue, and either person 2's eyes are blue or not blue. At each layer of hypothetical, the color of another person's eyes is lost, because "person 1 imagining person 2 imagining person 3" doesn't know the colors of any of those three people's eyes.
After the first night, everyone, including the hypothetical people, knows that there is more than one person with blue eyes. Person 1 knows that person 2 is no longer imagining person 3 imagining person 4 imagining person 5 imagining... ...person 98 imagining person 99 imagining that person 100 is the only blue eyed person.
Appendix: The entire expansion the recursion
Definitions:
- Person(N) - The n'th person on the island (ordered 1-100 without loss of generality).
- Hypo(N) - The n'th layer of imagination, starting from Person(N), assuming that every person in the chain imagines their own eyes to be non-blue. For example, in Hypo(1), Person(1) imagines that Person(2) sees 98 pairs of blue eyes.
Hypo(1): Person(1) imagines that Person(2) sees 98 pairs of blue eyes.
Hypo(2): Person(1) imagines that Person(2) imagines that Person(3) sees 97 pairs of blue eyes.
Hypo(3): Person(1) imagines that Person(2) imagines that Person(3) imagines that Person(4) sees 96 pairs of blue eyes.
...
Hypo(99): Person(1) imagines that Person(2) imagines that Person(3) imagines that Person(4) imagines that Person(5) imagines that Person(6) imagines that Person(7) imagines that Person(8) imagines that Person(9) imagines that Person(10) imagines that Person(11) imagines that Person(12) imagines that Person(13) imagines that Person(14) imagines that Person(15) imagines that Person(16) imagines that Person(17) imagines that Person(18) imagines that Person(19) imagines that Person(20) imagines that Person(21) imagines that Person(22) imagines that Person(23) imagines that Person(24) imagines that Person(25) imagines that Person(26) imagines that Person(27) imagines that Person(28) imagines that Person(29) imagines that Person(30) imagines that Person(31) imagines that Person(32) imagines that Person(33) imagines that Person(34) imagines that Person(35) imagines that Person(36) imagines that Person(37) imagines that Person(38) imagines that Person(39) imagines that Person(40) imagines that Person(41) imagines that Person(42) imagines that Person(43) imagines that Person(44) imagines that Person(45) imagines that Person(46) imagines that Person(47) imagines that Person(48) imagines that Person(49) imagines that Person(50) imagines that Person(51) imagines that Person(52) imagines that Person(53) imagines that Person(54) imagines that Person(55) imagines that Person(56) imagines that Person(57) imagines that Person(58) imagines that Person(59) imagines that Person(60) imagines that Person(61) imagines that Person(62) imagines that Person(63) imagines that Person(64) imagines that Person(65) imagines that Person(66) imagines that Person(67) imagines that Person(68) imagines that Person(69) imagines that Person(70) imagines that Person(71) imagines that Person(72) imagines that Person(73) imagines that Person(74) imagines that Person(75) imagines that Person(76) imagines that Person(77) imagines that Person(78) imagines that Person(79) imagines that Person(80) imagines that Person(81) imagines that Person(82) imagines that Person(83) imagines that Person(84) imagines that Person(85) imagines that Person(86) imagines that Person(87) imagines that Person(88) imagines that Person(89) imagines that Person(90) imagines that Person(91) imagines that Person(92) imagines that Person(93) imagines that Person(94) imagines that Person(95) imagines that Person(96) imagines that Person(97) imagines that Person(98) imagines that Person(99) imagines that Person(100) sees no people with blue eyes.
At each layer of the hypothetical, each hypothetical person is imagining a world where they do not have blue eyes. At 99-hypotheticals deep, a world is imagined (being imagined being imagined... being imagined) where only one person has blue eyes.