Mystery of Infinity-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?