Suppose that there exists $(a,b) \in S$ such that $f(x,y) \leq f(a,b)$ on $S$, then show that $(\Delta f) (a,b) \leq 0$
64764
Sign of Laplacian at critical points of $\mathbb R^n$
149739
Prove: $f(t)\leq t+1$
74950
Show a function for which $f(x + y) = f(x) + f(y) $ is continuous at zero if and only if it is continuous on $\mathbb R$
51659
If $f(0)=0$ then prove that $|f(z)+f(-z)| \leq 2|z|^2$
126821
Let $f:[0,1]\to \mathbb{R}$. Show that $\{a \in [0,1]: f(a)\neq 0\} = \bigcup_{n=1}^{\infty} \{a \in [0,1]: |f(a)|\geq 1/n\}$
76863
How to show $ \sup \inf g(x,y) \leq \inf \sup g(x,y)$?
129716
Let $f$ be continuous on $ [a,b]$, differentiable on $ (a,b) $ and $ f(x) \neq 0 \forall x \in (a,b) $ Prove $ \exists c \in (a,b) $ such that
85465
Let $f$ be increasing on $[a,b]$ and $a < x_1 < \dotsb < x_n < b$. Show that $\sum_{k = 1}^n [f(x_k^+) - f(x_k^-)]\leq f(b^-) - f(a^+)$.
59513
For $a, b \geq 0$, $0 < x < 1$, show $(a+b)^x \leq a^x + b^x$
29198
How to prove the following inequality $|\prod_{i=1}^{i=n}a_i-\prod_{i=1}^{i=n}b_i| < n\delta$?
151493
Using $\epsilon$-$\delta$ to show a property of a function
136789
How to prove $a \lt b$ for all $a \in A, b \in B$ implies $\sup A \leq \inf B$?
66951
Proving the existence of a sequence of functions converging to 0 with an added property
55461
How to prove that if $a$ belongs to $\mathbb R$, such that $0\leq a \leq\epsilon$, then $a = 0$
102478
$X_n\leq Y_n$ implies $\liminf X_n \leq \liminf Y_n$ and $\limsup X_n \leq \limsup Y_n$
105084
Let f be a function twice differentiable and with derivatives continuous on an interval $[a,b]$ containing $0$. Prove the following statement:
88064
Prove: $a_n \leq b_n \implies \limsup a_n \leq \limsup b_n$