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Mar
10
revised The equivalence between Cauchy integral and Riemann integral for bounded functions
deleted 16 characters in body
Mar
10
asked The equivalence between Cauchy integral and Riemann integral for bounded functions
Mar
2
comment limit of a sequence of functions equality
See if Arzela's dominated convergence theorem helps in my post.
Feb
28
comment Prove that $\lim\limits_{n\rightarrow \infty}\int_1^3\frac{nx^{99}+5}{x^3+nx^{66}} d x$ exists and evaluate it.
Since $(nx^{99}+5)/(x^3+nx^{66})$ is uniformly bounded on $[1,3]$, we can apply Lebesgue's dominated convergence theorem to interchange the $\lim$ and $\int$.
Feb
28
revised The limit of the measures of monotone decreasing sets
deleted 9 characters in body
Feb
23
comment The role of sequences in calculus
@BrianM.Scott I have a calculus book, Курс дифференциального и интегрального исчисления, translated from Russian, introducing Moore-Smith convergence without prerequisite of any topology knowledge, even metric spaces.
Feb
23
comment The role of sequences in calculus
I think Moore-Smith convergence might help.
Feb
23
accepted Does continuous convergence imply uniform convergence?
Feb
23
revised Does continuous convergence imply uniform convergence?
added 81 characters in body
Feb
23
comment Does continuous convergence imply uniform convergence?
@DavidMitra You're right. Could you pleased post an answer and I'll accept yours?
Feb
23
accepted Unnecessary simple function in the proof of Lebesgue's convergence theorem in Baby Rudin?
Feb
23
asked Does continuous convergence imply uniform convergence?
Feb
23
comment The limit of the measures of monotone decreasing sets
@Thomas If $\{A_n\}$ are measurable, it seems that $\mu(A_n)\to0$ since $A_n$ are bounded. We can write $A_1=\bigcup_n(A_n\backslash A_{n+1})$, since $\forall a\in A_1,\exists N: a\in A_N\land a\not\in A_{N+!}$, and $\mu(A_1)=\sum_n\mu(A_n\backslash A_{n+1})$. Since both sides are finite, we have $\mu(A_m)=\sum_{n\ge m}\mu(A_n\backslash A_{n+1})\to0$ as $m\to\infty$.
Feb
23
comment The limit of the measures of monotone decreasing sets
@Thomas The extension process is referred to Baby Rudin 11.7 to 11.10, just like the Lebesgue measure extended from the length function of elementary sets.
Feb
23
comment The limit of the measures of monotone decreasing sets
@Thomas From Baby Rudin: A nonnegative additive set function $\phi$ defined on $\mathcal E$ is said to be regular if the following is true: To every $A\in\mathcal E$ and to every $\epsilon>0$ there exists $F\in\mathcal E$, $G\in\mathcal E$ such that $F$ is closed, $G$ is open, $F\subset A\subset G$ and $\phi(G)-\epsilon\le\phi(A)\le\phi(F)+\epsilon$.
Feb
22
answered If $f_n(x_n) \to f(x)$ whenever $x_n \to x$, show that $f$ is continuous
Feb
22
comment A Limit of a sum related to the exponential series.
@SeanEberhard Yeah, for example, $k/(n+k)\le\ln(1+k/n)\le k/n$.
Feb
22
answered Asymptotic behaviour of a sequence
Feb
22
comment if $(n+1)a_{n+1}=(n-2)a_n+\frac{(n+1)^2(3n+2)}{4}(n\geq 2),~a_1=a_2=0 $ then $a_n=?$
@rigordonma I haven't gone into the calculation of $\sum_k (k+1)^2k(k-1)(3k+2)/4$.
Feb
22
answered A simultaneous system of equations