Someone said that 0.9=1 0. \overline{9} = 1

Calculus Level 2

Statement One: Let 0.49=x\text{Let} \ 0.4\overline{9} = x

Statement Two:     10x=4.9\implies 10x = 4.\overline{9}

Statement Three: 100x=49.9100x = 49.\overline{9}

Statement Four:     x=0.49=4590=12\implies x = 0.4\overline{9} = \cfrac{45}{90} = \cfrac{1}{2}

Since from Statement Four: , 0.49=4590=12 0.4\overline{9} = \cfrac{45}{90} = \cfrac{1}{2} , multiplying both sides by 2,

Statement Five:     [0.49]2=[12]2=1\implies \left[ 0.4\overline{9} \right ] \cdot 2 = \left[ \cfrac{1}{2} \right ] \cdot 2 = 1

Upon multiplying, we see that

Statement Six: [0.49]2=0.98=1 \left[ 0.4\overline{9} \right ] \cdot 2 = 0.\overline{9}8 = 1

Statement Seven: But upon searching, famously, 1=0.9 1 = 0.\overline{9}

Statement Eight: 0.98=0.9 \therefore 0.\overline{9}8 = 0.\overline{9}

But, it will break the "reflexive property of equality" .

Which Statement Is The Start Of The Mistake In The Said Argument?

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