I'm trying to generalize the following statement about Cauchy's Criterion:
A sequence in $\mathbb{R}$ satisfies CC $<=>$ it converges
And I want to generalize it to CC of a sequence in $\mathbb{R}^p$.
Here is the proof I have for CC in $\mathbb{R}$:
Assume the sequence converges, so $s_n \to s$ for some $s \in \mathbb{R}$. Since $|s_m - s_n| \leq |s_m - s| + |s - s_n|$, each term ont he right is $< \epsilon / 2$ for $m, n > N_{\epsilon / 2}$, so $|s_m - s_n| < \epsilon$, so CC is satisfied.
Assume CC, now we show $s_n \to s$ for some $s$. We can show that $CC$ implies that the sequence is bounded. Indeed, if we choose any $\epsilon > 0$, we can show that the sequence is bounded by $|s_n| \leq C$ with $C = \max(|s_1|, |s_2|, \dots, |s_{N_{\epsilon}}|, |s_{N_\epsilon + 1}+\epsilon)|$. Since it is bounded, some subsequence $s_{n_k} \to s$ for some $s$. By the triangle inequality, $|s_n - s| \leq |s_n - s_{n_k}| + |s_{n_k} - s|$. We can make each term on the right $< \epsilon / 2$ by Cauchy's Criterion and convergence, respectively. Thus $s_n \to s$.
When generalizing this to $\mathbb{R}^p$, if we make $s_n = (s_n^{(1)}, \dots, s_n^{(p)})$ a sequence in $\mathbb{R}^p$ instead of $\mathbb{R}$, is there anything that needs to be changed? I'm reading over the proof with this in mind and all of the logic still holds (assuming we have already proven the analogous theorems in $\mathbb{R}^p$, such as a bounded sequence in $\mathbb{R}^p$ has a convergent subsequence).
You can do it directly like what you had in mind or you can show that $(s_i^1,...,s_i^p)$ converges in $\mathbb{R}^p$ iff $s_i^j$ converge in $\mathbb{R}$ for $j=1..p$. Likewise for the Cauchy criterion. So you don't need a whole new proof, you can reduce it to the 1 dimensional case.
That is, your new proof would be $(s_i^1,...,s_i^p)$ converges in $\mathbb{R}^p$ iff $s_i^j$ converge in $\mathbb{R}$ for $j=1..p$ iff $s_i^j$ are Cauchy in $\mathbb{R}$ iff $(s_i^1,...,s_i^p)$ is Cauchy in $\mathbb{R}^p$