# Homework 4: Real Analysis

Problem 1

Let $$f: [a,b]\to\mathbb R$$ be a continuous function and $$(a_n)_{n=1}^{\infty}$$ a sequence of numbers from the interval $$[a,b]$$. Prove that there exists $$x\in[a,b]$$ such that $f(x)=\sum_{n=1}^{\infty} \frac{f(a_n)}{2^n}.$

Problem 2

If $$f:[a,b]\to\mathbb R$$ is a continuous function that is differentiable on $$(a,b)$$ prove that there exists $$c\in (0,1)$$ such that $f(1)-f(0)=\frac{f^{\prime}(c)+f^{\prime}(1-c)}2.$

Problem 3

Find the limit $\lim_{x\to 0}\frac{1-(\cos x)^{\sin x}}{x^3}.$

Problem 4

• (a) Assume that $$(a_n)_{n=0}^{\infty}$$ and $$(b_n)_{n=0}^{\infty}$$ are two sequences of integers. Prove the integration by parts formula for sequences: $\sum_{n=1}^k a_n(b_{n}-b_{n-1})=a_kb_k-a_0b_0-\sum_{n=0}^{k-1}b_n(a_{n+1}-a_n).$

• (b) Assume that $$(a_n)_{n=1}^{\infty}$$ and $$(b_n)_{n=1}^{\infty}$$ are two sequences such that $$(a_n)_{n=1}^{\infty}$$ is decreasing, positive, and with limit $$0$$, and $$(b_n)_{n=1}^{\infty}$$ is a sequence for which there exists $$M> 0$$ such that $$\left|b_1+b_2+\cdots+ b_k\right|< M$$ for all $$k\in\mathbb N$$. Prove that the series $$\displaystyle\sum_{n=1}^{\infty}a_nb_n$$ is convergent.

• (c) Prove that the series $$\displaystyle\sum_{n=1}^{\infty}\frac{\sin n}{\sqrt n}$$ is convergent.

Problem 5

Find the limit $\lim_{n\to+\infty}\left(\frac1{n+1}+\frac1{n+2}+\cdots+\frac1{5n}\right).$

Problem 6

Find the maximum value of $\int_0^1 \left(x^2f(x)-xf^2(x)\right)\,dx$ where the maximum is taken over all continuous functions $$f:[0,1]\to\mathbb R$$.

Problem 7

Assume that $$f:\mathbb R\to\mathbb R$$ is a convex function that satisfies $$\displaystyle\lim_{x\to+\infty}\frac{f(x)}{x}=+\infty$$. Prove that $\lim_{x\to+\infty}\left(f(2x)-2f(x)\right)=+\infty.$

Problem 8

Assume that $$f: \mathbb R_+\to\mathbb R_+$$ is a differentiable function such that for all $$x> 0$$ the following inequality is satisfied: $xf^{\prime\prime}(x)+f^{\prime}(x)+f(x)\leq 0.$ Prove that $$\displaystyle \lim_{x\to+\infty}f(x)=0$$.

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