Sure, let’s do it in base 3. 3 in base 3 is 10, and 3^(-1) is 10^(-1), so:

1/3 in base 10 = 1/10 in base 3 0.3… in base 10 = 0.1 in base 3

Multiply by 3 on both sides:

3 × 0.3… in base 10 = 10 × 0.1 in base 3 0.9… in base 10 = 1 in base 3.

But 1 in base 3 is also 1 in base 10, so:

0.9… in base 10 = 1 in base 10