This is not a proof as you start with the answer, albeit disguised as a known truth. Here is a real proof. Start by assigning the recurring decimal a variable.
x = 0.9999...
Now calculate 10 times this by shifting the decimal place.
10x = 9.9999...
You can then subtract the second equation from the first. Note that all the digits after the decimal cancel out, leaving us with the following.
9x = 9
x = 1
Therefore, 0.9999… = 1. Infinity does weird things!
This is not a proof as you start with the answer, albeit disguised as a known truth. Here is a real proof. Start by assigning the recurring decimal a variable.
Now calculate 10 times this by shifting the decimal place.
You can then subtract the second equation from the first. Note that all the digits after the decimal cancel out, leaving us with the following.
Therefore, 0.9999… = 1. Infinity does weird things!