Find the limit of the sequence $\{.1, .101, .101001, \ldots,\}$.

Question

Reference:- Foundations of Functional Analysis, S. Ponnusammy, $\alpha-science-2002$

Consider the sequence $\{.1, .101, .101001, \ldots\}$, prove that given sequence is Cauchy and converges to a irrational number. find the limit of the sequence.

I tried to find the generalised formula for the sequence pattern. I failed to do. $\left\{\frac{1}{10}, \frac{1}{10}+\frac{1}{10^3}, \frac{1}{10}+\frac{1}{10^3}+\frac{1}{10^6}, \ldots\right\}$. It doesn't form any geometric series. How do I prove the sequence is Cauchy without getting the rigourous expression for the $n$-th term of the given sequence? how to find the limit of the sequence?

Answer 1

You can prove it is Cauchy directly from the definition. If I give you an $\epsilon \gt 0$ you just need to find an $N$ such that no two terms beyond $N$ differ by more than $\epsilon$. All the changes after $k$ add up to less than $10^{-k}$

Answer 2

Hint: If $a$ and $b$ are positive, smaller than $1$, and they agree up to $k$ decimal places, then $\lvert a-b\rvert<\ldots$