Position of specific value

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Let's assume a have an arbitrarily long number, take π for example. Since we know π is infinite, there will at some point be a group of numbers like "2015201620172018...", correct? My question is, for any arbitrarily long number, is there a mathematical way of determining the position of a specified set of numbers in said long number? The only conceivable way I can think of is similar to how we find the coefficient of x using the binomial theorem, but that leaves the problem of determining the number in the format of a binomial expansion.

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A string being infinite does not necessarily mean any finite string is included in it. Take for example the decimal expansion

$$\frac{1}{3}=0.333\dots$$

Clearly there are an infinite number of decimals, but that does not mean that you can find any finite string of numbers you want in there. Furthermore, the answer to your question is that no, there is no way to do so for any arbitrary number which satisfies the conditions you need. Indeed even proving that a number has such a property is an extremely difficult thing to do to begin with.