are p-adic values infinite if using a limit?

Question

I watched a Veritasium video about p-adic numbers.

In 2:30 they take the 10-adic number ...857142857143, they multiply it by 7 and it happens that this number is designed in a way that all products except the first one result in a number ending in 0 and carrying some value.

Since the 10-adic number has infinite digits, this results in a 1 with infinite zeros on the left, so in the video they then claim that this equals 1, and therefore this 10-adic number represents the number 1/7

...857142857143
   x          7
   ____________
...000000000001

Saying that the result is 1 makes some kind of logic (we have a non ending amount of zeros on the left, so what else could it be?)

However there's always a non-zero carry that will never disappear that we just completely discarded without any more explanation.

Here is my logic:

Let's start with a finite multiplications using the first 4 digits of the previous example:

  7143
  x  7
 _____
 50001      
 

The result is 50001

We can write this result as a sum of two terms: the 1 that we obtain if we discard the last carry, and the carry itself, which will be some value k multiplied by 10^i where i is the index at which the carry finally settles (in this case 4)

50001 = 5*10^4 + 1

Now if instead of a finite 4 digit number we have a number with infinite numbers, we just take the limit when i tends to infinity and we calculate the result.

Here k represents the carry value, which we know it is a non-zero integer in the range [1..9]

$$\lim_{i\to\infty} k\cdot10^i + 1$$

The result of this sum (that does take in account the carry and doesn't magically discard it as they do in the video) is, as expected, infinite, meaning that the number ...857142857143 represents an infinite value.

Is there any flaw in my logic of using a limit to take in account the value of the carry, and therefore obtaining an infinite value instead of a value of 1?

I understand that such number could never be constructed, so their infinite value could never be achieved. But the same applies to functions that converge to a value, we know that they will never reach that value but we have tools like limits to compute what that value is.

I know I must be wrong, since p-adic numbers is an field of mathematics, and the fact that they represent finite numbers is accepted, I just want to see what's wrong with the limit approach that sets its value to infinite.