We all know the famous identity:

\(0.9999\ldots{ }\ldots{ }\ldots = 1\).

Here the numbers on the both sides are in base \(10\).

One special thing here about \(9\) is: in this case, \(9\) just one less than the base \(10\).

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

The equality \(0.ccc\ldots{ }\ldots{ }\ldots = 1\) is true in every base \(b \geq 2\), where \(c\) is the largest digit in base \(b\), equivalently, \(c=b-1\).

Is this conjecture **TRUE**?

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