Mystery of Infinity-2

Number Theory Level 2

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?