I'm not going to be able to produce a full proof here but take the case of a number going, say, 3.141 ...357 357 357 ... where after some finite length of digits the pattern finally emerges and you get a repeating loop.
Now multiply this be 1,000, and you would get 3141. ... 357 357 357 ...
Subtracting the two gives say 3138. ... Note that there are as many 357's at the end as there were before, because there is an infinite number of them. And more importantly all of these cancel each other so that now our decimal expansion comes to an end somewhere.
So we have the formula: (1000 -1)*pi = 999 pi = 3138. ... Coming to a stop. You could then write pi in the form (3138. ... ) / 999. This can become a fraction if you multiply top and bottom by enough ten's to move all of the decimal places to the left of the decimal point.
So, eventually, you can find a finite, closed form of pi. This argument could extend to any repeating pattern at the end of pi, of any length. And the point is that this leads to an equation that says that pi = a/b, for possibly stupendously big but nevertheless finite numbers. And it has been proved that such a form does not exist for pi, so that there is no repeating patter of any length, anyway, for inifinitely long.
Yes, at some points patterns emerge, e.g there is a 01234567890 somewhere in the string. But no, these patterns are never more than one-offs, and the cycle of digits goes on and on, never repeating itself overall. Proven fact.