Bijection invalidating cantor's diagonalization argument?

Question

I am a software engineer without a math degree, so I am planning to learn something today.

Take this bijection between the naturals and reals. (This is a valid bijection, no?)

...03020 => 0.02030... 
...11111 => 0.11111... 
...51413 => 0.31415...
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.
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Walking along the diagonal, we can find a real number not listed. However, that would also find a natural number not listed as well, correct?

Answer 1

All natural numbers have finitely many digits.

Answer 2

Forget diagonals, the point of being an bijection is that you can go both ways. If your map is a bijection, it has an inverse that sends reals to naturals. But where would this inverse send the famous real number $1/3 = 0.33333333...$?