Intereting Posts

$n$ points are picked uniformly and independently on the unit circle. What is the probability the convex hull does not contain the origin?
How can I find a homeomorphism from $\mathbb{R}^n$ to the open unit ball centered at 0?
Is every submodule of a projective module projective?
Solve $y^2= x^3 − 33$ in integers
Convergence of Martingale.
Transfinite derivatives
boundary of the boundary of a set is empty
Details about Cayley's Group Theorem
Prime ideals in $\mathbb{Z}$
Rate of convergence of series of squared prime reciprocals
Number of Derangements of the word BOTTLE
What equation produces this curve?
Number of pairs of strings satisfying the given condition
Compact spaces and closed sets (finite intersection property)
Norms on C inducing the same topology as the sup norm

This is a question from Bergman’s companion to Rudin.

a) Show that the only polynomials which are bounded as functions $\mathbb{R} \rightarrow \mathbb{R}$ are constant functions.

(I can do this) Also done here

- If a separately continuous function $f : ^2 \to \mathbb{R}$ vanishes on a dense set, must it vanish on the whole set?
- If $\sum_{n=1}^{\infty} a_{n}^{3}$ converges does $\sum_{n=1}^{\infty} \frac{a_{n}}{n}$ converge?
- Limit Supremum and Infimum. Struggling the concept
- Tricks. If $\{x_n\}$ converges, then Cesaro Mean converges (S.A. pp 50 2.3.11)
- If $\{x_{2m}\}$ and $\{x_{2m-1}\}$ converge to the same limit, does $\{x_m\}$ converge?
- Too simple proof for convergence of $\sum_n a_n b_n$?

b)Deduce that if a sequence of polynomials $P_n:\mathbb{R} \rightarrow \mathbb{R}$ converges uniformly on $\mathbb{R}$ to $f$ then $f$ is a polynomial.

I figure that the uniform convergence implies at some point (for large n) the polynomials must have the same highest power because otherwise large values of $\mathbb{R}$ would destroy any hope of uniform convergence. Then eventually the second highest power must be equal as well by a similar argument…Then I guess you could make a similar argument for the co-efficients by plugging in large values of x, the difference in each co-efficient must be quite small in order to maintain the uniform convergence.

I would like some help understanding if/why this means that the limit actually is a polynomial.

- Sufficiency to prove the convergence of a sequence using even and odd terms
- Show that $(f_n)$ is equicontinuous, given uniform convergence
- Uniform convergence of series involving sin x: $\sum_{n=1}^\infty \frac{\sin x}{1+n^2x^2}$
- How to show that $f'(x)<2f(x)$
- $\lim\limits_{x\to\infty}f(x)^{1/x}$ where $f(x)=\sum\limits_{k=0}^{\infty}\cfrac{x^{a_k}}{a_k!}$.
- How do I calculate $\lim_{x\rightarrow 0} x\ln x$
- Why do we say “radius” of convergence?
- $f:\mathbb R \to \mathbb R$ is continuous and lim$_{n\to \infty} f(nx)=0$ for all real $x$ $\implies $ lim$_{x \to \infty}f(x)=0$
- Does convergence in $L^{p}$ implies convergence almost everywhere?
- Evaluating $\int_{0}^{1}\frac{1-x}{1+x}\frac{\mathrm dx}{\ln x}$

Hint: if $f_n$ converges uniformly, there exists $n$ such that $|f_n – f_m| \le 1$ for all $m \ge n$.

- Examples of faithfully flat modules
- Changing the values of an integrable function $f: \to \mathbb R$ countably infinitely many points not a dense subset of $$
- Can anyone sketch an outline of Iwaniec's proof for the upper bound regarding the Jacobsthal function?
- Non-Abelian group $G$ in which $x\mapsto x^3$ is a homomorphism
- Integers that satisfy $a^3= b^2 + 4$
- Let $L$ be a Lie algebra. why if $L$ be supersolvable then $L'=$ is nilpotent.
- Metric space is totally bounded iff every sequence has Cauchy subsequence
- Converging series in Banach space
- If $A^2\succ B^2$, then necessarily $A\succ B$
- Why do we need to check for more than $\frac{\infty}{\infty}$ or $\frac{0}{0}$ when applying L'Hospital?
- Constructing a Galois extension field with Galois group $S_n$
- moment generating function
- Proving that there is a bijection between $A$ and $B$.
- Presentation of Rubik's Cube group
- Can the real numbers be forced to have arbitrary cardinality?