Intereting Posts

Given $P(B\mid A)=1-\varepsilon$ for some $0<\varepsilon<1$ and $P(C\mid B)=1$, prove that $P(C\mid A)≥1-\varepsilon$
An application of Yoneda Lemma
Show that any solution of second order differential equation has atmost a countable number of zeroes $?$
Closed-form for rational power sum
In written mathematics, is $f(x)$a function or a number?
Polynomial splitting in linear factors modulo a prime ideal
Is the geometric dot product formula equal to the algebraic one and how can I get one from the other in a step by step fashion ?
What is the Coxeter diagram for?
Names of higher-order derivatives
Minimize $\|AXBd -c \|^2$, enforcing $X$ to be a diagonal block matrix
How to find k term in a series.
Find a function $f: \mathbb{R} \to \mathbb{R}$ that is continuous at precisely one point?
Sums of independent Poisson random variables
Let $u:\mathbb{C}\rightarrow\mathbb{R}$ be a harmonic function such that $0\leq f(z)$ for all $ z \in \mathbb{C}$. Prove that $u$ is constant.
A proof of uniformly continuous function and bounded set

If $\lim a_n = L$, then $\lim s_n = L$, where $s_n = \frac{ a_1 + \dots + a_n}{n}$

I know that:

$|s_n – L| = |\frac{a_1 + \dots + a_n}{n} – L| = \frac{1}{n}|a_1 + \dots + a_n – nL| \leq \frac{1}{n}\sum_{i=1}^{n}|a_i -L|$

- Evaluate $\int_{-\infty}^\infty x\exp(-x^2/2)\sin(\xi x)\ \mathrm dx$
- The Biharmonic Eigenvalue Problem with Dirichlet Boundary Conditions on a Rectangle
- An explicit construction for an everywhere discontinuous real function with $F((a+b)/2)\leq(F(a)+F(b))/2$?
- The preimage of continuous function on a closed set is closed.
- Can the proof of Theorem 1.20 (b) in the book, The Principles of Mathematical Analysis by Walter Rudin, 3rd ed., be improved?
- isomorphism of Dedekind complete ordered fields

- for infinite compact set $X$ the closed unit ball of $C(X)$ will not be compact
- To find the minimum of $\int_0^1 (f''(x))^2dx$
- How to prove and what are the necessary hypothesis to prove that $\frac{f(x+te_i)-f(x)}{t}\to\frac{\partial f}{\partial x_i}(x)$ uniformly?
- Analog of $(a+b)^2 \leq 2(a^2 + b^2)$
- Using integral definition to solve this integral
- Differentiability-Related Condition that Implies Continuity
- Does absolute convergence of a sum imply uniform convergence?
- Bounded derivative implies uniform continuity on an open interval
- At large times, $\sin(\omega t)$ tends to zero?
- If the derivative approaches zero then the limit exists

Let $\varepsilon > 0$. You know that for some $N$, $|a_n – L| < \varepsilon/2$ for $n > N$.

We also know that there exists some $K$ such that $|a_n – L| < K$ for all $n$. For example, you can take $K = 1 + \text{max}(|a_1 – L|, \cdots, |a_N – L|,\varepsilon/2)$.

Now take any $m > N$. We have:

$$|s_m – L| \leq \frac{1}{m}\sum_{i = 1}^m|a_i – L| = \frac{1}{m}\left(\sum_{i = 1}^N|a_i – L| + \sum_{i = N + 1}^{m}|a_i – L|\right) < \frac{1}{m}(NK + (m – N)\frac{\varepsilon}{2}) < \frac{NK}{m} + \frac{\varepsilon}{2}$$

Now just take $M \geq \text{max}(N,\frac{2NK}{\varepsilon})$. So, for $m > M$, $\frac{NK}{m} < \frac{\varepsilon}{2}$, which implies $|s_m – L| < \varepsilon$.

let $\epsilon>0$ has given.$$\exists N\in\mathbb{N}~~ \text{s.t.}~~~ n > M, |a_n − L| <\frac{\epsilon}{2}$$ so

$$|s_n-l| = \left|\frac{ a_1 + \dots + a_n}{n}-l\right|= \left|\frac{ a_1 + \dots + a_n-nl}{n}\right|= \left|\frac{ a_1-l + \dots+a_N-l}{n}\right| + \left|\frac{a_N+l+\dots + a_n-l}{n}\right| < \left|\frac{ a_1-l + \dots+a_N-l}{n}\right| +\frac{(n-N)\epsilon}{2n} $$ suppose $c=a_1-l + \dots+a_N-l$ thus first fraction=$\frac{c}{n}\to 0$ this will give you $N_2>N ~~\text{s.t.}~~\left|\frac{c}{n}\right|<\frac{\epsilon}{2}~~~ \forall n>N_2$

- The p-adic numbers as an ordered group
- The prime number theorem and the nth prime
- Let $Z$ be standard normal can we find a pdf of $(Z_1,Z_2)$ where $Z_1=Z \cdot 1_{S},Z_2=Z \cdot 1_{S^c}$
- Why is the 3D case so rich?
- Generators for $S_n$
- Equivalence of definitions of multivariable differentiability
- How can I prove $dz=dx+idy$?
- Treatise on non-elementary integrable functions
- Combinatorial proof involving factorials
- Integral Basis for Cubic Fields
- Prove $G$ is a group (unusual star operation).
- Why this two surfaces have one end?
- Positive integer multiples of an irrational mod 1 are dense
- Is is possible to obtain exactly 16 black cells?
- Calculating probability with n choose k