Prove that $\sup[f(x)+g(x)]\le \sup(f(x)) + \sup (g(x))$
10859
Prove $\sup(f+g) \le \sup f + \sup g$
56413
$f(g(x))=g(f(x))$ implies $f(c)=g(c)$ for some $c$
153272
Basic question about $\sup_{x\neq 0}{} \frac{\|Ax\|}{\|x\|} = \sup_{\|x\| = 1}{\|Ax\|} $, $x \in\mathbb{R}^n$
139111
Prove that $a<c<b$, $\frac{f(b)-f(a)-(b-a)f'(a)}{g(b)-g(a)-(b-a)g'(a)}=\frac{f"(c)}{g"(c)}$
24856
Proving sup(A + B) = sup A + sup B
6450
Show $\mathbb{E}[f(X)g(X)] \geq \mathbb{E}[f(X)]\mathbb{E}[g(X)]$ for $f,g$ bounded, nondecreasing
13253
Prove $f(S \cup T) = f(S) \cup f(T)$
149634
$ 2|f^{'}(0)| = \sup_{z, w \in D} |f(z)-f(w)|$
121264
Prove that rank $(g \circ f)\le \min[\operatorname{rank}(f),\operatorname{rank}(g)]$; $g$ and $f$ are linear mappings
69921
Prove that: $(1+a_1)(1+a_2)...(1+a_n) \le 1 + S_n + (S_n)^2/2! + ... + (S_n)^n/n!$
97772
An example of a function which satisfies $\sup_{y}\inf_{x}f(x,y)<\inf_{x}\sup_{y}f(x,y)$
4498
Proof that $\|fx\| \leq \|f\|\cdot\|x\|$
104044
Prove continuous functions $f$ and $g$ are equal
145423
prove that if $1 \le m \le n$ then $\left(1+\frac1n\right)^m < 1+\frac mn + \left(\frac mn\right)^2$
43478
Prove that: $ m_a + m_b + m_c \le 4R + r $
58943
If $f'(x)\le g'(x)$, prove $f(x)\le g(x)$
81775
Prove $a^\alpha b^{1-\alpha} \le \alpha a + (1 - \alpha)b, \; a,b > 0,\; 0 < \alpha < 1$