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 Nov 6 revised For a closed plane curve, showing some inequalities. removed an incorrect remark and corrected a link Nov 4 comment Properties of a smooth bijection For $M=N=\Bbb R$, $x\mapsto x^3$ is not of constant rank. Nov 4 awarded Revival Nov 1 revised prove $f(x)$ has at least $2n$ roots deleted 160 characters in body Nov 1 answered prove $f(x)$ has at least $2n$ roots Oct 27 comment Limit of $x_n/n$ for sequences of the form $x_{n+1}=x_n+1/x_n^p$ @sundaycat: To take limit in $(5)$, I need assumption (ii) to apply $(1)$, where a reasonable requirement is $\theta_n$ cannot be too large. Without any assumption, I only know $0<\theta_n<\delta_n$. To control the upper bound of $\theta_n$, I need assumption $(1)$ to obtain $\delta_n\le\delta$ when $n$ is large. For example, consider $f(x)=x^p$($p\ge 1$) and an arbitrary $\theta:(0,\infty)\to (0,\infty)$. Then you need some condition on $\theta$ to get $\lim_{x\to\infty}\frac{f'(x+\theta(x))}{f'(x)}=1$. Oct 26 comment Limit of $x_n/n$ for sequences of the form $x_{n+1}=x_n+1/x_n^p$ @MartinArgerami: Thanks for your comment. I completely changed my argument after reading other answers. Oct 26 revised Computing a limit almost surely using the strong law of large numbers added 39 characters in body; edited title Oct 26 awarded Yearling Oct 26 revised Limit of $x_n/n$ for sequences of the form $x_{n+1}=x_n+1/x_n^p$ added 76 characters in body Oct 25 comment Limit of $x_n/n$ for sequences of the form $x_{n+1}=x_n+1/x_n^p$ From my answer you can see that letting $y_n=x_n^2$ is a good choice for your question (1) but not as good a choice for your question (2). The reason for letting $y_n=x_n^2$ in question (1) can be understood in this way: the associated function $f$ is question (1) is $c x^2$ for some $c>0$. For the same reason, in question (2), a good choice is letting $y_n=x_n^{3/2}$. Oct 25 revised Limit of $x_n/n$ for sequences of the form $x_{n+1}=x_n+1/x_n^p$ added 2340 characters in body Oct 23 revised Limit of $x_n/n$ for sequences of the form $x_{n+1}=x_n+1/x_n^p$ added 141 characters in body Oct 23 answered Limit of $x_n/n$ for sequences of the form $x_{n+1}=x_n+1/x_n^p$ Oct 22 reviewed Leave Open Calculating index of a subgroup Oct 22 comment Is the image of a nowhere dense closed subset of $[0,1]$ under a differentiable map still nowhere dense? @NateEldredge: Here $E$ is closed and hence.... Oct 22 comment A Problem on Improper Integrals @ParamanandSingh: Thank you for your tolerance. Oct 22 comment A Problem on Improper Integrals @ParamanandSingh: You are welcome! I still feel sorry about misleading you with a flawed argument. Maybe I I shouldn't post any answer when my head is not clear enough. Don't worry about the bounty; you can wait until the bounty expires to see if there is a better answer. Oct 22 comment A Problem on Improper Integrals @ziangchen: Thank you again for your appreciation. Oct 22 comment A Problem on Improper Integrals @ParamanandSingh: Terribly sorry! I mistakenly thought $h(0)=0$ before. I have updated my answer.