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
Do you know what an irrational number is?