User Incompl33t - Mathematics Stack Exchange

8 votes

1 answer

466 views

Show that $\lim_{x \to a}{P(x)} = P(a)$ for any polynomial $P(x)$

asked Aug 1, 2022 at 19:59

7 votes

1 answer

85 views

Solution verification: Usage of L'Hôpital's rule, derivatives of trigonometric functions

asked May 10, 2022 at 19:57

5 votes

4 answers

719 views

Explanation for a solution: Howard Anton, Elementary Linear Algebra

asked Apr 25, 2022 at 20:11

4 votes

2 answers

88 views

Solution verification: showing $\lim \limits_{n \to \infty} \frac{n!}{n^n} = 0$

asked Apr 21, 2022 at 19:38

4 votes

1 answer

81 views

Let $0 < a \le b$. Then $a^2 \le \left (\frac{2ab}{a+b} \right) ^2 \le ab \le \left( \frac{a+b}{2} \right) ^2 \le b^2$

asked Jan 9, 2023 at 14:45

3 votes

1 answer

69 views

Axler LADR, ex. 2.A.16

asked Feb 2 at 7:42

3 votes

3 answers

141 views

Find $\int \sin^4x dx$

asked Jun 20, 2022 at 19:42

2 votes

2 answers

392 views

Inverse of a strictly increasing function

asked Mar 27, 2022 at 9:46

2 votes

2 answers

86 views

Definition of the Logarithm Function

asked May 25, 2022 at 17:59

2 votes

2 answers

73 views

Proof that $\binom{n}{k}\frac{1}{n^k} \le \frac{1}{k!}$

asked Dec 11, 2022 at 6:12

2 votes

4 answers

83 views

Proof that $\vert a - b \vert \lt c \iff -c \lt a - b \lt c$

asked Jun 4, 2022 at 13:06

2 votes

1 answer

20 views

Inequality in a proof that every b-adic number is a Cauchy-Sequence

asked Feb 12, 2023 at 16:07

1 vote

0 answers

67 views

Sheldon Axler LADR Exercise 3.D.4

asked May 21, 2023 at 12:56

1 vote

4 answers

85 views

Proof of $B \subseteq A^C \iff A \cap B = \emptyset$

asked Jul 27, 2022 at 21:16

1 vote

2 answers

80 views

If $B_1, B_2 \subseteq Y$ and $f: X \rightarrow Y$ then $f^{-1}(B_1 \cap B_2)=f^{-1}(B_1) \cap f^{-1}(B_2)$

asked Jan 20, 2023 at 12:13

1 vote

3 answers

168 views

Derivative of $g(x) = \frac{\log x}{\log a}$

asked May 25, 2022 at 18:12

1 vote

1 answer

430 views

Proof that $(A^t)^t=A$, $(A+B)^t=A^t+B^t$, $(AB)^t=B^tA^t$, and deduce that $BB^t$ is symmetric and $B-B^t$ is skew-symmetric

asked May 17, 2022 at 20:43

1 vote

2 answers

44 views

Integration of a piecewise defined discontinuous function

asked Apr 17, 2022 at 11:53

1 vote

2 answers

195 views

Proof for Squeeze Theorem for Sequences

asked Jun 4, 2022 at 12:46

1 vote

3 answers

46 views

$2a-1< x+a < 2a+1 \implies |x+a| < 2|a|+1$

asked Aug 6, 2022 at 16:48

1 vote

0 answers

106 views

Proof for the multiplication law of limits

asked Aug 9, 2022 at 7:25

1 vote

1 answer

82 views

Continuity of $f$ doesn't imply differentiability

asked Aug 16, 2022 at 6:04

0 votes

0 answers

160 views

If $\sum a_n$ is divergent then $\sum -a_n$ also diverges

asked Jul 9, 2022 at 11:45

0 votes

1 answer

81 views

Calculus by Spivak, Excercise 13-7

asked Apr 13, 2022 at 12:47

0 votes

2 answers

116 views

Prove $\lim_{x\to 0^+} f\left(1/x\right) = \lim_{x\to \infty} f(x)$

asked May 31, 2022 at 21:11

0 votes

0 answers

85 views

If $\lim_{n\to\infty} a_n=a$, then $\lim_{n\to\infty}\sum_{k=0}^n\frac{a_k}{n+1}=a$

asked Jan 31, 2023 at 9:56

0 votes

1 answer

36 views

Equivalence of Dedekind cuts and Dedekind left sets

asked Feb 8, 2023 at 16:58

0 votes

0 answers

29 views

If $f$ is a continuous mapping from a metric space $X$ to a metric space $Y$ and $E$ a dense subset of $X$. Is$f(X) = f(\bar{E})$?

asked May 7 at 15:05

-4 votes

4 answers

135 views

Prove that $A \cap B = A \cap C$ and $A \cup B = A \cup C$ if and only if $B = C$

asked Jul 27, 2022 at 20:57

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29 Questions

Show that $\lim_{x \to a}{P(x)} = P(a)$ for any polynomial $P(x)$

Solution verification: Usage of L'Hôpital's rule, derivatives of trigonometric functions

Explanation for a solution: Howard Anton, Elementary Linear Algebra

Solution verification: showing $\lim \limits_{n \to \infty} \frac{n!}{n^n} = 0$

Let $0 < a \le b$. Then $a^2 \le \left (\frac{2ab}{a+b} \right) ^2 \le ab \le \left( \frac{a+b}{2} \right) ^2 \le b^2$

Axler LADR, ex. 2.A.16

Find $\int \sin^4x dx$

Inverse of a strictly increasing function

Definition of the Logarithm Function

Proof that $\binom{n}{k}\frac{1}{n^k} \le \frac{1}{k!}$

Proof that $\vert a - b \vert \lt c \iff -c \lt a - b \lt c$

Inequality in a proof that every b-adic number is a Cauchy-Sequence

Sheldon Axler LADR Exercise 3.D.4

Proof of $B \subseteq A^C \iff A \cap B = \emptyset$

If $B_1, B_2 \subseteq Y$ and $f: X \rightarrow Y$ then $f^{-1}(B_1 \cap B_2)=f^{-1}(B_1) \cap f^{-1}(B_2)$

Derivative of $g(x) = \frac{\log x}{\log a}$

Proof that $(A^t)^t=A$, $(A+B)^t=A^t+B^t$, $(AB)^t=B^tA^t$, and deduce that $BB^t$ is symmetric and $B-B^t$ is skew-symmetric

Integration of a piecewise defined discontinuous function

Proof for Squeeze Theorem for Sequences

$2a-1< x+a < 2a+1 \implies |x+a| < 2|a|+1$

Proof for the multiplication law of limits

Continuity of $f$ doesn't imply differentiability

If $\sum a_n$ is divergent then $\sum -a_n$ also diverges

Calculus by Spivak, Excercise 13-7

Prove $\lim_{x\to 0^+} f\left(1/x\right) = \lim_{x\to \infty} f(x)$

If $\lim_{n\to\infty} a_n=a$, then $\lim_{n\to\infty}\sum_{k=0}^n\frac{a_k}{n+1}=a$

Equivalence of Dedekind cuts and Dedekind left sets

If $f$ is a continuous mapping from a metric space $X$ to a metric space $Y$ and $E$ a dense subset of $X$. Is$f(X) = f(\bar{E})$?

Prove that $A \cap B = A \cap C$ and $A \cup B = A \cup C$ if and only if $B = C$