To give a concrete example for you, lets start with the finite set: {0.22, 0.32, 0.33} We now construct a new number in this way: First digit of the first n

To give a concrete example for you, lets start with the finite set: {0.22, 0.32, 0.33} We now construct a new number in this way: First digit of the first number is 2, so we take 3 which is different as our first digit. First digit of the second number is 3 so we take 2 as our second number. Now we have 0.32 First digit of the third number is 3 so we take 0 as our third number. We've constructed the number: 0.320 = 0.32 which is in the set already. So this construction method doesn't guarantee that it's a new number, only that it's different from the first. With Cantor's Diagonalization argument we guarantee it's different from the first number since it's different in the first digit, then we guarantee it's different from the second since it's different in the second digit, etc. etc. it's different from all the numbers in our list. To take the same set above as an example, we end up creating a number like: 0.318 which is definitely different from the numbers in the set