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SomeStrangeUser
  • Member for 10 years, 11 months
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23 votes

Prove that every convex function is continuous

16 votes
Accepted

If $\lambda$ is an eigenvalue of $A^2$, then either $\sqrt{\lambda}$ or $-\sqrt{\lambda}$ is an eigenvalue of $A$

10 votes
Accepted

Representing $\frac{3 + \sqrt{5}}{2}$ as a square of a quadratic surd

9 votes
Accepted

The solutions of $(z-i)^n+(z+i)^n=z^n$ are real.

9 votes
Accepted

The functional equation $\frac{f(x)}{f(1-x)} = \frac{1-x}{x}$

6 votes
Accepted

Proving a limit with a logarithm exists

4 votes
Accepted

Evaluating the limits $\lim_{(x,y)\to(\infty,\infty)}\frac{2x-y}{x^2-xy+y^2}$ and $\lim_{(x,y)\to(\infty,8)}(1+\frac{1}{3x})^\frac{x^2}{x+y}$

4 votes

Multiples of an irrational number forming a dense subset

3 votes

Let $f: \mathbb R^k \rightarrow \mathbb R$ continuous and unbounded in both directions. Show $f(\mathbb R^k)=\mathbb R$

3 votes
Accepted

If $f(g(x)) = 4x^2-8x$ and $f(x)=x^2-4$, then what's the value of $g(x)$?

3 votes

Why $\mu(\lim \sup(A_n)) = 0$ if $\sum_{n\geq 1} \mu(A_n)<\infty$?

3 votes
Accepted

Check that if F(X)=aX then DF(X)=aI

3 votes
Accepted

How to prove that ${n\choose 0} < {n\choose 1} < \ldots < {n\choose (n+1)/2}$

2 votes

Is there a direct way to prove $\lim\limits_{n\to\infty}(1+\frac{1}{n})^n=\sum\limits_{n=0}\frac{1}{n!}$?

2 votes

Finding the probability of $y \leq \sin x.$

2 votes
Accepted

Finding a closed form for $\sum\limits_{\substack{0\le n\le N\\0\le m\le M}}\left|nM-Nm\right|$

2 votes
Accepted

General formula of $\lim_{n \to \infty} (1+a_n)^{b_n}$ where $b_n \to \infty$

2 votes

Wrong basic exponential rule: $(a^b)^c\neq a^{bc}$

2 votes
Accepted

Inverse of $I+BA$ is $(I+BA)^{-1} = I-B(I+AB)^{-1}A$

2 votes
Accepted

How to do this last step in this proof that inner product preserving implies linear?

2 votes

Show that $f(x)\equiv 0$ if $ \int_0^1x^nf(x)\,dx=0$

2 votes

Suppose the gcd (a,b) = 1 and c divides a + b. Prove that gcd (a,c) = 1 = gcd (b,c)

2 votes
Accepted

What is the probability of choosing r objects from c different groups when there are m groups of n objects?

2 votes
Accepted

Disjoint Exhaustive Subsets with Alternate Elements and equal Cardinality

1 vote

Prove: $\int_0^{\infty}\left(\frac{\sin x}{x}\right)^2dx=\pi/2$

1 vote

Evaluate the given limit in $C_r=\{re^{i\theta}:0\le \theta \le \pi\}$

1 vote

Prove the inequality $0<(m+n)/(mn-1)\le 3$ for $m,n\in\mathbb N $ with $mn\ne 1$

1 vote

Prove $f_{n}(x)=x^{n}-x^{2n}$ for all $x\in[0,1]$ point convergence

1 vote

Prove $\lim_{x\to\infty}\frac{x^{x^2}}{2^{2^x}}$

1 vote
Accepted

accumulation point of a sequence