Mystery of Infinity-2

We all know the famous identity:

0.9999=10.9999\ldots{ }\ldots{ }\ldots = 1.

Here the numbers on the both sides are in base 1010.

One special thing here about 99 is: in this case, 99 just one less than the base 1010.

So, noticing this special property, one can make the conjecture that

The equality 0.ccc=10.ccc\ldots{ }\ldots{ }\ldots = 1 is true in every base b2b \geq 2, where cc is the largest digit in base bb, equivalently, c=b1c=b-1.

Is this conjecture TRUE?


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