Tricky functions and how they function!

Relevant wiki: Is 0.999... = 1?

A simple proof that $0.999 \dots = 1$ is shown below:

PROOF

$x = 0.999 \dots$ then,

$10x = 9.999 \dots$ ,

If we subtract $x$ from both sides then,

$9x = 9$

Divide both sides by $9$

$x = 1 = 0.999 \dots$

And thus $\lfloor 1 \rfloor = 1$

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You can read more about it in the wiki

$\sum_{k=0}^{\infty}0.9(0.1)^k=0.9\frac{1}{1-0.1}=1$ and the integer part is $\boxed{1}$ .

Im writing this post because I'm a bit confused. Let's formulate the problem as below.

$lim_{n\rightarrow \infty} \lfloor 9\sum_{k=1}^{n}(0.1)^k\rfloor$

I do not think the limit of this expression is equal to $1$ (it looks more like $0$ ). However, if we write as below, then the answer is $1$

$\lfloor lim_{n\rightarrow \infty} 9\sum_{k=1}^{n}(0.1)^k\rfloor$

I do not know which one is correct, so, any comment is welcome.