Articles of real analysis

Drawing large rectangle under curve

Let $f$ be a continuous nonincreasing function on $[0,1]$ with $f(1)=0$ and $\int_0^1 f(x)dx=1$. Does there exist a constant $k$ for which we can always draw a rectangle with area at least $k$, with sides parallel to the axes, in the area bounded by the two axes and the curve $f$? If we choose the […]

For any subset $S$ of $\mathbb{R}$, the distance function $d_s(x) = d(x,S)$ is continuous

Let $S \subset \mathbb R$ be any set, and define for any $x \in \mathbb R$ the distance between $x$ and the set $S$ by $d(x,S) = \inf\{|x-s| : s \in S\}$. Prove that the function $d_s: \mathbb R \to [0,+\infty)$ given by $d_s(x) = d(x,S)$, is Lipschitz continuous. Prove that if $S$ is compact […]

Using Ratio test/Comparison test

I have $\displaystyle\sum_{n=1}^{\infty}{\frac{(2n)!}{(4^n)(n!)^2(n^2)}}$ and need to show whether it diverges or converges. I attempted to use the ratio test, but derived that the limit of $\dfrac{a_{n+1}}{a_n}=1$ and hence the test is inconclusive. So I now must attempt to use the comparison test, but I am struggling to find bounds to compare it to to show […]

$L_p$ complete for $p<1$

It is rather straight forward to show that $L_p$ is complete for $p\geqslant 1$, but I am having trouble showing the same thing when $p<1$. For the former case I have shown that every absolutely convergent sequence converges by constructing a a function in $L_p$ but bigger than the series and used the dominated convergence […]

are singletons always closed?

I am learning about metric spaces and I find it very confusing. Is this a valid proof that a singleton must be closed? If $(X,d)$ is a metric space, to show that $\{a\}$ is closed, let’s show that $X \setminus \{a\}$ is open. Choose $y \in X \setminus \{a\}$ and set $\epsilon = d(a,y)$. Then […]

Proof that in a metric space $X$, if $\phi \in \mathbb{R}^X$ is not continuous, then $\{ x \ | \ \phi(x) \geq \alpha \}$ is not necessarily closed

In the last few days I already posted two alternative proofs (here and the other available link) of the basic result in metric spaces that, given a continuous function $\phi \in \mathbb{R}^X$, the set $\{ x \ | \ \phi(x) \geq \alpha \}$ is closed for any $\alpha \in \mathbb{R}$. As explained there, I am […]

Verifying that Lipschitz functions carry measure 0 sets to measure 0 sets

Let $f: [a,b] \rightarrow \mathbb{R}$ be a Lipschitz function. Let $\epsilon > 0$. Let $E$ be a set of measure zero. There exists countable, bounded, open intervals with the form $I_n=(x_n, y_n)$ such that $E \subseteq \bigcup\limits_n I_n$ and for some $x_k, y_k \in I_k$, then $|f(x_k)-f(y_k)| \leq c|x_k -y_k|$. I believe this sufficiently incorporates […]

Taylor expansion of composite function

My confusion is slightly related to this question. Suppose we have two nice functions $f(x)$ and $g(x)$, how do we find Taylor series of $f(g(x))$? To be more concrete, consider $f(x^2)$. In this case, we can regard it as $f(g(x))$ where $g(x) = x^2$. One way to find the Taylor series around $1$ is just […]

Convergence in distribution ( Two equivalent definitions)

I read that for convergence in distribution it is equivalent to have that either the characteristic functions of the random variables convergence pointwise or we have that $F_{X_n} \rightarrow F_{X}$ pointwise, where $F$(the distribution function) is continuous. I could not find a proof of this, so I was wondering how hard it is to show? […]

Find all continuous functions $f(x)^2=x^2$

Find all functions $f$ which are continuous on $\mathbb R$ and which satisfy the equation $f(x)^2=x^2$ for all $x \in \mathbb R$. Clearly $f(x)=x, -x, |x|, -|x|$ all satisfy the condition. However, how can I show that these must be the only possible choices? The condition guarantees that $|f(x)|=|x|$, for all $x$ so I think […]