I am currently self-studying Hubbard's textbook on Vector Calculus but am stuck on the logic for this question:
0.4.1(a): "Let $x$ and $y$ be two positive reals. Show that $x + y$ is well defined by showing that for any $k$, the digit in the kth position of $[x]_N + [y]_N$ is the same for all sufficiently large $N$. Note that $N$ cannot depend just on k, but must depend also on $x$ and $y$."
Here $[x]_N = \ ...x_{N-1}x_N\ 0\ 0\ 0\ ...$ means the decimal expansion of $x$ is truncated at its Nth digit $x_N$.
My initial thought is that we take $N$ to be $\ \inf S\ $ where $S = \{n\ |\ x_n+y_n \geq 10\ and\ n>k \}\ $ if $S \neq \varnothing$ otherewise we take $N=k$
However I'm not sure if this is correct or how I might prove this is correct. Any help with solving the problem would be appreciated.
The sequences $N\mapsto[x]_N$ and $N\mapsto[y]_N$ are increasing sequences, and the same is true for the sequence $N\mapsto s_N:=[x]_N+[y]_N\in{\mathbb Q}$. Assume that a $k$ is given. Then the sequence $N\mapsto[s_N]_k$ is increasing as well. Since the values are in a bounded discrete set of rational numbers the $[s_N]_k$ $(N\to\infty)$ have to be constant after some time.