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 Apr19 accepted How can I show that there are only finitely many solutions for the following system? Aug25 comment How can I show that there are only finitely many solutions for the following system? sorry for the tag and thanks for the hint. Aug25 asked How can I show that there are only finitely many solutions for the following system? Jul22 awarded Enthusiast Jul15 awarded Scholar Jul15 accepted Does there exist a set in the plane such that topological dimension 2 with empty interior? Jul15 accepted How to show that this limit $\lim_{n\rightarrow\infty}\sum_{k=1}^n(\frac{1}{k}-\frac{1}{2^k})$ is divergent? Jul15 accepted Find the limit $\displaystyle\lim_{n\rightarrow\infty}{(1+1/n)^{n^2}e^{-n}}$? Jul15 accepted How to show that $\sqrt{1+\sqrt{2+\sqrt{3+\cdots+\sqrt{2006}}}}<2$ Jul15 revised find angle in triangle added 24 characters in body Jul15 awarded Teacher Jul15 answered find angle in triangle Jul3 comment How to show that this limit $\lim_{n\rightarrow\infty}\sum_{k=1}^n(\frac{1}{k}-\frac{1}{2^k})$ is divergent? @julien, thanks. Jul3 comment How to show that this limit $\lim_{n\rightarrow\infty}\sum_{k=1}^n(\frac{1}{k}-\frac{1}{2^k})$ is divergent? So, Is this enough to say this series is divergent? Jul3 asked How to show that this limit $\lim_{n\rightarrow\infty}\sum_{k=1}^n(\frac{1}{k}-\frac{1}{2^k})$ is divergent? Jun30 awarded Informed Jun30 awarded Commentator Jun28 comment Prove that $\sqrt{7}^{\sqrt{8}}>\sqrt{8}^{\sqrt{7}}$ $\sqrt{7}^{16}>\sqrt{8}^{14}$. So if we take $\frac{1}{2}$ power of both sides, we have $\sqrt{7}^{8}>\sqrt{8}^{7}$. Maybe i am wrong. Jun28 comment Prove that $\sqrt{7}^{\sqrt{8}}>\sqrt{8}^{\sqrt{7}}$ It is true that $7^8>8^7$, since $(1+\frac{1}{7})^7\sqrt{8}^7$. Does it imply that $\sqrt{7}^\sqrt{8}>\sqrt{8}^\sqrt{7}$. Jun17 comment $\frac{c}{1+c}\leq\frac{a}{1+a}+\frac{b}{1+b}$ for $0\leq c\leq a+b$ You can also use the monotone increasing property of $\frac{x}{1+x}$ for proving $\frac{d(x,y)}{1+d(x,y)}$ is a metric.