query: Prove $\omega + \omega_1 = \omega_1$
positive_passages.docid: 25498
positive_passages.text: Prove that $\omega + \omega_1 = \omega \cdot \omega_1 = \omega^{\omega_1} = \omega_1$
negative_passages.docid: 145632
negative_passages.text: Proving $\frac{1}{\cos^2\alpha}<\frac{\tan\beta-\tan\alpha}{\beta-\alpha}<\frac{1}{\cos^2\beta}$
negative_passages.docid: 134768
negative_passages.text: Borel hierarchy doesn't "collapse" before $\omega_1$
negative_passages.docid: 101778
negative_passages.text: Show that there is $\eta\in(0,1)$ with $f''(\eta)=f(\eta)$
negative_passages.docid: 10702
negative_passages.text: Show that $\left\{A\subset\Omega: A~\text{ is countable or }A^C\text{ is countable}\right\}$ is a $\sigma$-algebra
negative_passages.docid: 129813
negative_passages.text: Is there an easy proof for ${\aleph_\omega} ^ {\aleph_1} = {2}^{\aleph_1}\cdot{\aleph_\omega}^{\aleph_0}$?
negative_passages.docid: 79686
negative_passages.text: How can I prove these statements?
negative_passages.docid: 15262
negative_passages.text: Prove $\exists \xi, \eta \in (0,1)$, such that $f(\xi)+f'(\eta)=\xi+\eta$.
negative_passages.docid: 149822
negative_passages.text: If $\omega \in \Omega^q(M)$, and $(\omega|_{U})_p = 0$, is $\omega_p = 0$?
negative_passages.docid: 91000
negative_passages.text: Prove that $\frac{1}{a+1}0$
negative_passages.docid: 21881
negative_passages.text: Prove that $\forall\alpha\geq\omega$, $|L_\alpha|=|\alpha|$ without AC
negative_passages.docid: 9003
negative_passages.text: prove $\frac{1}{2}\mathbf{\nabla (u \cdot u) = u \times (\nabla \times u ) + (u \cdot \nabla)u}$ using index notation
negative_passages.docid: 83185
negative_passages.text: How to show countability of $\omega^\omega$ or $\epsilon_0$ in ZF?
negative_passages.docid: 32965
negative_passages.text: Proving that for infinite $\kappa$, $|[\kappa]^\lambda|=\kappa^\lambda$
negative_passages.docid: 100608
negative_passages.text: Prove that $\|v\|_{H^1(\Omega )}\leq C(\|f\|_{L^2(\Omega )}+\|v\|_{H^{1/2}(\partial \Omega )}+\|\partial _\nu v\|_{H^{1/2}(\partial \Omega )}))$
negative_passages.docid: 89175
negative_passages.text: Proof of $(a+b)^{n+1}$
negative_passages.docid: 42849
negative_passages.text: Is it true that $\omega=\{0,1,2,3,\ldots\}$ in ZFC?