Show that there exists an integer $L<m≤ K$ such that $m/n$ is an upper bound for $E$, but that $(m−1)/n$ is not an upper bound for $E$.
11056
Terence Tao, Analysis I, Ex. 5.5.2: Entry point needed
103732
Proving that for $n \equiv 0 \pmod{2}$, we get ${n \choose 0} < {n \choose 1} <\ldots< {n \choose n/2-1}<{n\choose n/2}$ etc.
118195
How does one prove that $e$ exists?
7974
lim sup inequality $\limsup ( a_n b_n ) \leq \limsup a_n \limsup b_n $
56818
Prove that if $A$ is positive, then $\exists\;\alpha>0$ such that $\langle Ax, x \rangle\geq \alpha|x|^2$ for $x\in\Bbb{R}^n$
45914
Proving that $n!≤((n+1)/2)^n$ by induction
117262
Proving a logarithm inequality
118244
An $n$th root inequality: $\sqrt[n]{n} < 1 + \sqrt{2/n}$
2708
Inequality ${n \choose k} \leq \left(\frac{en}{ k}\right)^k$
143067
Show given any $x\in\mathbb{R}$ show there exists a unique $n\in\mathbb{Z}$ such that $n-1\leq x <n$.
43246
Showing that if $a<b$, then $a^{1/n} < b^{1/n}$
19804
How to prove that $|w_{n+1}-w_n| \leq (2/3)^2 |w_n - w_{n-1}|$?
124456
Given $x \in \mathbb{R}$ and $N \in \mathbb{N}, N>1,$ find integers $h,k$ with $0 < k \leq N $ that satisfy $\lvert{kx - h\rvert} < 1/N.$
39943
Proving $\mathrm e <3$
117640
Prove that $\lim \inf \dfrac{x_{n+1}}{x_n}\leq \lim \inf x_n^{1\over n} \leq \lim \sup x_n^{1\over n} \leq \lim \sup \dfrac{x_{n+1}}{x_n}. $
72600
Proof that $n!\leq {(\frac{n+1}{2})}^{n}$
19823
Inequality involving $\limsup$ and $\liminf$: $ \liminf(a_{n+1}/a_n) \le \liminf((a_n)^{(1/n)}) \le \limsup((a_n)^{(1/n)}) \le \limsup(a_{n+1}/a_n)$