tag:blogger.com,1999:blog-9004452969563620551.post6486293161765035515..comments2019-05-15T12:04:42.529+05:30Comments on Paramanand's Math Notes: The Riemann Integral: Part 1Unknownnoreply@blogger.comBlogger2125tag:blogger.com,1999:blog-9004452969563620551.post-45954856435249451332017-05-22T17:31:56.671+05:302017-05-22T17:31:56.671+05:30@Unknown, A sequence $a_{n}$ is said to be Cauch...@Unknown,<br /><br />A sequence $a_{n}$ is said to be Cauchy if for any $\epsilon&gt;0$ we have a positive integer $N$ such that $|a_{m} - a_{n} |&lt;\epsilon$ whenever $m\geq N, n\geq N$.<br /><br />The sequence $a_{n} =S(P_{n}, f)$ satisfies a different condition namely that for every positive integer $n$ we have $|a_{m} - a_{n} |&lt;1/n$ for $m&gt;n$. It is easy to show that if this condition is satisfied then the condition mentioned in definition of Cauchy sequence is also satisfied. Let $\epsilon&gt;0$ be given. Consider $n=[2/\epsilon] +1$ where $[x]$ represents greatest integer not exceeding $x$. This means that $n$ is a positive integer and $1/n&lt;\epsilon/2$. We choose $N=n+1$ and then if $r, s$ are positive integers with $r\geq N, s\geq N$ then both $r, s$ are greater than $n$ and hence by the specific property of $a_{n} =S(P_{n}, f)$ we have $|a_{r} - a_{n} |&lt;1/n&lt;\epsilon/2$ and a similar inequality holds for $a_{s}$. Now by triangle inequality it is easy to observe that $$|a_{r} - a_{s} |\leq |a_{r} - a_{n} |+|a_{n} - a_{s} |&lt;\frac{\epsilon} {2}+\frac{\epsilon}{2}=\epsilon$$ and thus we have shown that the sequence $a_{n}$ is Cauchy. Paramanand Singhhttps://www.blogger.com/profile/03855838138519730072noreply@blogger.comtag:blogger.com,1999:blog-9004452969563620551.post-53916401560736976882017-05-15T06:21:54.769+05:302017-05-15T06:21:54.769+05:30Forgive my ignorance but I remember that Cauchy se...Forgive my ignorance but I remember that Cauchy sequence was &quot;Exist N, such that if m&gt;N,n&gt;N then |Am-An|&lt;1/N&quot; ¿is the same as you say &quot;m&gt;n then |am-an|&lt;1/n?...thank very much...the rest of the note is very very good ...excellent..Unknownhttps://www.blogger.com/profile/03689274376715128595noreply@blogger.com