Let $S=\{\frac{1}{n}-\frac{1}{m}:n,m\in \mathbb{N}\}$ Show that $\sup S = 1$
96931
Supremum and infimum of $\{\frac{1}{n}-\frac{1}{m}:m, n \in \mathbb{N}\}$
108298
Prove $\sum_{i=1}^{i=n}\frac{a_i}{S-a_i}\geqslant \frac{n}{n-1}$ and $\sum_{i=1}^{i=n}\frac{S-a_i}{a_i}\geqslant n(n-1)$.
80350
$\limsup \frac{a_n}{b_n}0$ s.t $a_n\le Mb_n$
94425
What is $\sup(\sin(n))$?
141837
Proof of $\sup\{ (n!)^{(\frac{1}{n})}| n ∈ N\} = +∞?$
91081
Prove that, for every natural n, $\sum_{d|n} \frac{\mu(d)}{d}=\prod_{p|n}(1-\frac{1}{p})$
86505
Let $A,B \subset \mathbb{R}$. Show that $\sup(A \cup B) = \max\{\sup A, \sup B\}$
138107
Showing that $\left|\frac{z-a}{1-\bar{a}z}\right|=1$
153272
Basic question about $\sup_{x\neq 0}{} \frac{\|Ax\|}{\|x\|} = \sup_{\|x\| = 1}{\|Ax\|} $, $x \in\mathbb{R}^n$
14197
show that $\sum_{i=0}^l \dfrac{{l\choose i}}{{n\choose k + i}} = \dfrac{n+1}{(n-l+1) {n-l\choose k}}$
80277
How to show that $\lim \frac{1}{n} \sum_{i=1}^n \frac{1}{i}=0 $?
96152
Showing that $a^n - 1 \mid a^m - 1 \iff n \mid m$
13696
Proving $\limsup\frac 1 {a_n}=\frac 1 {\liminf a_n}$ and $\limsup a_n\cdot \limsup \frac 1 {a_n} \ge 1$
94558
Show that $\phi(mn) = \phi(m)\phi(n)\frac{d}{\phi(d)}$
87941
Prove by mathematical induction that $\forall n \in \mathbb{N} : \sum_{k=1}^{2n} \frac{(-1)^{k+1}}{k} = \sum_{k=n+1}^{2n} \frac{1}{k} $
62382
I suppose $(a,m)=(a-1,m)=1$, show that $1+a+a^2+\ldots+a^{\phi(m)-1}\equiv0\pmod m$
13291
Proving $\sum_{k=1}^n\frac1{\sqrt k}<2\sqrt n$ by induction