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### In space $C[0,1]$ find distance from point $x(t)=t^{2}+1$ to the subspace $L_{1}=\{x\in C[0,1]: x(0)=0\}$

I'am pretty sure that (a) is meant the following way: $C[0,1]$ is endowed with the maximum norm $\|\cdot\|_\infty$. $L_1$ is a subspace of $(C[0,1],\|\cdot\|_\infty)$, thus $L_1$ carries the maximum ...
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If it if unspecified, the usual distance between two elements $x,y \in C[0,1]$ is $$\| x-y\| = \sup_{t\in [0,1]} \vert x(t)- y(t)\vert.$$ The distance from an element $x \in C[0,1]$ to a subset $Y\... • 14.7k 2 votes ### Prove$|\prod_{i=1}^n a_i - a_n^n|\leq 2n\delta$if$0 \leq a_i \leq 1$and$|a_i - a_{i+1}| \leq \delta$The proposition is false. Take$a_1 = \frac{1}{2},\ a_2 = \frac{1}{2},\ n=2, \delta = 0,$so that the condition$|a_i - a_{i+1}| \leq \delta$for all$i<2$is satisfied. Then, $$\Big|\prod_{i=1}^n ... 2 votes Accepted ### Problem in proving that the set of integers n_k:k\geq1 is not bounded If n_k is bounded, then because \dfrac{m_k}{n_k} \in (0, 1) there are only finitely many possibilities for r_k=\dfrac {m_k}{n_k}. That means there are only finitely many differences between the ... • 17.6k 0 votes Accepted ### Verification of proof of convergence of the series \displaystyle \sum_{n \in \mathbb{N}} (-1)^n \dfrac{n!}{n^n} Your proof is fine. You may also notice that for any n\geq 1$$ \frac{n!}{n^n} = \int_{0}^{+\infty} n z^n e^{-nz}\,dz $$so$$ \sum_{n\geq 0}(-1)^n\frac{n!}{n^n} x^n = 1-x\int_{0}^{+\infty}\frac{z e^... • 343k 2 votes Accepted ### Is there a standard rule for$\arctansubstitutions? It's by a simple factorization (and experience). We have \begin{align} \int\frac{du}{a^2u^2+b^2}&=\int\frac{du}{b^2\left(1+\left(\frac{au}{b}\right)^2\right)} \end{align} This immediately suggests ... • 37.8k 1 vote Accepted ### Contraction Mapping Principal inC^1[a,b]Let $$F(f)(x) = 1 + \frac15\int_{0}^{x}\sin (tf(t))\, \mathrm dt$$ \begin{align} \left\|F(f) - F(g)\right\| &= |F(f)(0) - F(g)(0)| + \sup\limits_{t\in \left[0,\frac\pi2\right]} \left|F(f)'(t) - F(... • 5,549 1 vote Accepted ###\binom{n}{k}p^kq^{n-k}\sim\exp(-x^2_k/2)/\sqrt{2\pi npq})$,$x_k=(k-np)/\sqrt{npq}$. Be careful$e^{ab} \neq e^ae^b$(I think that you did the mistake in your last transition). However if you write$a_n = 1 + b_n$when$b_n \to 0\$, $$\log\phi(n,k) = \left(1+b_n\right) \left(-\frac{x_k^... • 5,549 0 votes ### A function with two center of symmetry - Prove that there does not exist an irrational number q such that f(ql-x)=f(ql+x). Proposition (2) is false. Suppose l=1. If either one of \alpha,\beta is irrational, then the proposition is false since you can take q=\alpha,\beta. It is the case for f(x)=\cos{\left(2\pi(x-\... • 1,147 2 votes Accepted ### Is this proof of monotone convergence theorem circular? The usual proof of the MCT uses the fact that the monotone limit of a sequence of nonnegative simple functions satisfies the limit-integral swaparoo property. I believe this is what is being used to ... • 4,584 0 votes ### Function defined through integral not continuous Let \varepsilon > 0 and define h(y)= \max\limits_{x \in (-\varepsilon, \varepsilon)} f(x,y) the minimal dominating function. We see that for any y, the maximum of f(x,y) is reached for x = ... • 315 0 votes ### What's the derivative of:  \sqrt{x+\sqrt{{x}+\sqrt{x+\cdots}}}? Clarification to solve the question The question posed is calcule Derivative \sqrt{x+\sqrt{x+\sqrt{x+....}}} we put y=\sqrt{x+\sqrt{x+\sqrt{x+....}}}  Before calculating the derivative we ... 0 votes ### Misunderstanding of the definition of an almost upper bound Hint: given z, how can z = x hold for infinitely many x? • 43k 1 vote ### Misunderstanding of the definition of an almost upper bound A is a set, so by definition it contains just one instance of any of its elements. There can't be "infinitely many x \in A" all equal to z. There is at most one. • 81.5k 2 votes Accepted ### Let \varphi_n, \psi be simple functions such that (\varphi_n) is increasing and \psi\le\lim_n\varphi_n. Then \int\psi\le\lim_n \int \varphi_n If you know the monotone convergence theorem, this is simple: By the monotone convergence theorem, we have \lim_{n \to \infty} \int \varphi_n = \int \lim _{n \to \infty}\varphi_n. Since \psi \leq \... • 8,158 1 vote Accepted ### Bounding super exponential functions with factorial functions Let's take logarithms of both sides, we then want to have p(n) \cdot \log_2 q(n)! > 2^n. We can use very rough estimation, k! < k^k, to get that left side is less than p(n) \cdot q(n)\cdot \... • 8,901 6 votes Accepted ### Borel Cantelli-Type problem from Billingsley's Probability and Measure It seems to be simpler to show the equivalence of the negation of the assertions, that is, we will prove that$$ \mathbb P\left(\limsup_{n\to\infty}A_n\right)<1\Leftrightarrow \mbox{ there exists }...
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