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I am studying Real Analysis on my own. I came across this proof of Cauchy's general principle of convergence. I cannot understand the logic behind the line indicated in blue: "Therefore there exists a natural number $q>m$ such that $|u_q-l|<\frac{\epsilon}{3}$. Why is $q$ greater than $m$? Any help would be appreciated.

  • $\begingroup$ The inequality holds for infinitely many values of $q$. How many integers $q$ are less than or equal to $m$? $\endgroup$ – Kavi Rama Murthy Sep 16 at 11:13

Note that a sequence is a mapping $f:\Bbb N\to \Bbb R$, and we write $x_n:=f(n)$. Now, for any strictly increasing function $i:\Bbb N\to \Bbb N$, the composition map $f\circ i:\Bbb N\to \Bbb R$ is called a subsequence and we denote $x_{i_k}:=f\big(i(k)\big)$.

Now, $l$ is a sub sequential limit of a sequence $\{x_n\}_{n\in \Bbb N}$ if there is a subsequence $\{x_{i_k}\}_{k\in \Bbb N}$ converging to $l$.

Hence, for any fixed positive integer $m$ we have $i(m)+1>m$ as $i$ is strictly increasing.

Also for given any $\epsilon'>0$ as $\lim_{k\to \infty}x_{i_k}=l$ we have $k_0\in \Bbb N$ such that $k\geq k_0$ implies $\big|x_{i_k}-l\big|<\epsilon'$. Hence, considering $p:=\max\{k_0,m\}+1$ and $q:=i(p)$ we are done.

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