Is it not enough that ⅓ = 0.333... ?

My thinking is, you start with a few intuitive understandings, like this:

• 1 ÷ 3 = ⅓
• ⅓ × 3 = 1

Layer in decimal long division and long multiplication:

• [Decimal long division] 1 ÷ 3 = 1 ⟌3 = 0.333...

• [Decimal long multiplication] 0.333... × 3 = 0.999...

It's convenient that 3 x 3 < 10 for this example, so there's no carryover placeholders when running the standard long multiplication algorithm, so you can run it from left to right, or in any order. Each 3 becomes a 9:

.333333...

x 3

------

.999999...

If all of that is satisfying, can't we then substitute in the answer we got, and .999... = 1? Help me understand what I'm missing.

Thanks!