# Questions tagged [pointwise-convergence]

For questions about pointwise convergence, a common mode of convergence in which a sequence of functions converges to a particular function. This tag should be used with the tag [convergence].

416 questions
Filter by
Sorted by
Tagged with
796 views

### Theorems similar to Dini's Theorem and Egoroff's Theorem

Dini's Theorem states that Given a sequence of real-valued continuous functions $(f_n)$ on a compact set $E\subseteq \mathbb{R},$ if $(f_n)$ decreases to a continuous function $f$ pointwise on $E$, ...
742 views

### Counterexample around Dini's Theorem

"Give an example of an increasing sequence $(f_n)$ of bounded continuous functions from $(0,1]$ to $\mathbb{R}$ which converge pointwise but not uniformly to a bounded continuous function $f$ and ...
86 views

### Pointwise convergence of sequence $(f_n)_n$ of functions to $f$ and changing limits

My analysis notes contains the following question: if $(f_n)_n$ is a sequence of functions of $A \subset \mathbb{R} \to \mathbb{R}$ and $a \in \mathbb{R} \cup \{-\infty, +\infty\}$ an accumulation ...
133 views

### Does $f_{n}(z) = \frac{1}{1+n^{2}|z-e^{in}|}$ converge pointwise or uniformly?

I want to check whether the sequence below converges pointwise or uniformly $$f_{n}(z):\{z\in\mathbb{C}:|z| = 1\}\to\mathbb{R}$$ $$\qquad\qquad f_{n}(z) = \frac{1}{1+n^{2}|z-e^{in}|}$$ I have tried ...
140 views

### Criteria for smoothness of the pointwise limit of a sequence of functions

Let $\#$ denote cardinality, and fix $p\in[1,\infty]$. Let $(f_n)_{n\in\mathbb{N}}$ be a sequence of functions from $[0,1]$ to $\mathbb{R}$ with the properties $f_n\in C^{n-1}([0,1])$ $f^{(n)}_n$ is ...
108 views

178 views

### The limit of a pointwise converging sequence of polynomial is smooth

Let $(P_n)_{n\geqslant 0}$ being a sequence of real polynomials with non negative coefficients, that converge pointwise on $\mathbb{R}$ to a functoin $f$. Then: $f\in \mathcal{C}^{\infty}$. How ...
2k views

### What is the difference between pointwise boundedness and boundedness?

In my textbook, the Arzela-Ascoli Theorem states that for a compact set $A$ in a metric space $M$ and $B\subset\mathcal{C}\left(A, N\right)$, where $N$ is another metric space, then $B$ is compact if ...
42 views

### Will we have uniform convergence in this case?

Assume that $f: [a,b] \rightarrow \mathbb{R}$ is continuously differentiable. Then we know that for each $t \in [a,b]$ we have that $$\frac{f(t+\Delta t)-f(t)}{\Delta t}-f'(t)$$ will converge to ...
32 views

### Prove that $\lim\limits_{n\rightarrow \infty}P_n(A)=P(A)$ implies $\lim\limits_{n\rightarrow \infty} \int f ~dP_n = \int f ~dP$

Let $P_n, n \in \mathbb{N}$ and $P$ be probability measures on the measurable space $(\Omega,\mathfrak{S})$ and assume $\forall A \in \mathfrak{S}: \lim\limits_{n\rightarrow \infty}P_n(A)=P(A)$. I ...
145 views

33 views

### Proving this pointwise limit via definition

For $f_n(x) = \frac{nx + x^2}{n^2}$, this converges pointwise to the function $f(x) = 0$. How would I prove this formally? This is my attempt: |f_n(x) - f(x)| = |\frac{nx + x^2}{n^2}| \leq |\frac{...
70 views

### Pointwise convergence of $f_n(x) = (x+1)\arctan(x^n)$

I want to study the pointwise convergence of $f_n(x) = (x+1)\arctan(x^n)$ on $R$ but I have trouble establishing pointwise convergence on the interval $I = [-\infty, -1)$. My reasoning is the ...
57 views

45 views

### Sequence of functions that converges strongly in $L^2(\mathbb R)$ but not pointwise

I am trying to find a sequence of functions $\{f_j\}$ and a function $f$ such that $f_j\to f$ strongly in $L^2(\mathbb R)$ but $f_j$ does not converge to $f$ pointwise. The definition of strong ...
26 views

### Proof Verification for Uniform Convergence on Sequence of Functions

just looking for a verification on a proof. Thanks in Advance Let $f_n$ be a sequence of functions such that $f_n=\frac{x^{2n}}{1+x^{2n}}$ defined on $[-2,2]$. Prove or Disprove Uniform Convergence ...
109 views

38 views

### Finding the Pointwise Limit of a Function

If I have a sequence of functions $f_n[0,2] \rightarrow \mathbb{R}$ where $f_n(x) = \frac{x^n}{2^n+n}$. If I attempt to find the pointwise limit, I work out that by taking $x \in [0,2]$: We can ...
### Prove the compactness and second countability of a subset of $\Bbb{R}^{L^1(G)\times D}$
I'm trying to fulfill the details in a proof whose aim is to guarantee the existence of a compact space with certain properties, but in this case I had no success. Let $X$ be a metric space (actually,...
Let $(X,\|\cdot\|)$ be a (infinite dimensional) Banach space and $f_{n}:[0,1]\longrightarrow (X,\|\cdot\|)$ continuous, for each integer $n\geq 1$, a sequence of mappings. There is some "pointwise ...