List questions from solution-verification

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Proof question, inequalities

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$S_n$ is a group

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Alternative proof to Theorem 2.37 in Baby Rudin

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Prove that the upper integral $\int_{-1}^1f(x)dx$ is equal to $0$.

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Solution of Problem 22 in section 4 in chapter 12 of Royden (real analysis third edition (2))

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Let $I=[a,b]$ and $f:I\to \mathbb{R}$ continuous. $f(x)>0$ for $x\in I$. Prove $\exists~\alpha>0$ s.t. $f(x)>\alpha~\forall~x\in I$.

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Let $A,B$ be sets. Show that the three statements $A\subseteq B$, $A\cup B = B$ and $A\cap B = A$ are logically equivalent.

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Let $A,B,C$ be sets. Show that $A\subseteq C$ and $B\subseteq C$ if and only if $A\cup B \subseteq C$

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Is this $2 \times 2$ matrix a subspace

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Problems 11.3, No. 14 (a) and (b) from page 237 of David M. Burton's "Elementary Number Theory" (7th Edition)

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$T_1$-property of cofinite topology.

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Understanding details in proof of: Suppose $X,$ compact Hausdorff and $Y$ Hausdorff such that $X \subseteq Y$. Then, $X$ is closed in $Y$

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Where do these conditional probabilities come from?

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What is the problem with this differential equation?

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List Question

Limit of a bounded increasing sequence

Proof question, inequalities

$S_n$ is a group

Alternative proof to Theorem 2.37 in Baby Rudin

Prove that the upper integral $\int_{-1}^1f(x)dx$ is equal to $0$.

Solution of Problem 22 in section 4 in chapter 12 of Royden (real analysis third edition (2))

Let $I=[a,b]$ and $f:I\to \mathbb{R}$ continuous. $f(x)>0$ for $x\in I$. Prove $\exists~\alpha>0$ s.t. $f(x)>\alpha~\forall~x\in I$.

Let $A,B$ be sets. Show that the three statements $A\subseteq B$, $A\cup B = B$ and $A\cap B = A$ are logically equivalent.

Let $A,B,C$ be sets. Show that $A\subseteq C$ and $B\subseteq C$ if and only if $A\cup B \subseteq C$

Is this $2 \times 2$ matrix a subspace

Problems 11.3, No. 14 (a) and (b) from page 237 of David M. Burton's "Elementary Number Theory" (7th Edition)

$T_1$-property of cofinite topology.

Understanding details in proof of: Suppose $X,$ compact Hausdorff and $Y$ Hausdorff such that $X \subseteq Y$. Then, $X$ is closed in $Y$

Where do these conditional probabilities come from?

What is the problem with this differential equation?

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