Intereting Posts

Finding $A,B\in SL_2(\Bbb{Z})$ of finite order with the property that $AB=C$ where the order of $C$ is infinite.
True or False $A – C = B – C $ if and only if $A \cup C = B \cup C$
How can we show that an abelian group of order <1024 has a set of generators of cardinality <10
Removing the star without changing homology
Evaluate the integration : $\int\sqrt{\frac{(1-\sin x)(2-\sin x)}{(1+\sin x)(2+\sin x)}}dx$
sign of the conditional expectation
$\int\limits_{a}^{b} f(x) dx = b \cdot f(b) – a \cdot f(a) – \int\limits_{f(a)}^{f(b)} f^{-1}(x) dx$ proof
Floyd's algorithm for the shortest paths…challenging
Elliptic Regularity on Manifolds
Condition for a ring on projective and free modules problem
Number theory problem, 3rd degree diophantine equation
Three dimensional spherical excess formula
Method of proof of $\sum\limits_{n=1}^{\infty}\tfrac{\coth n\pi}{n^7}=\tfrac{19}{56700}\pi^7$
Characterising Continuous functions
Prove that $\frac{\partial x}{\partial y} \frac{\partial y}{\partial z} \frac{\partial z}{\partial x} = -1$ and verify ideal gas law

Let $X$ and $Y$ metric spaces, $f$ is an injective from $X$ to $Y$, and $f$ sets every compact set in $X$ to compact set in $Y$. How to prove $f$ is continuous map?

Any comments and advice will be appreciated.

- Uniform continuity and translation invariance
- Need Help: Any good textbook in undergrad multi-variable analysis/calculus?
- Positivity of the Coulomb energy in 2d
- Why do we need min to choose $\delta$?
- Riemann integral question
- Proof of the Strenghtened Limit Comparison Test

- Type of singularity of $\log z$ at $z=0$
- How to prove convex+concave=affine?
- A type of local minimum (2)
- Infinite dimensional integral inequality
- Show a bounded linear operator is weakly sequentially continuous
- Convergence of series involving iterated $ \sin $
- Meaning of “kernel”
- Dirac delta distribution & integration against locally integrable function
- Inverse Function Theorem for Banach Spaces
- Let $(s_n)$ be a sequence of nonnegative numbers, and $\sigma_n=\frac{1}{n}(s_1+s_2+\cdots +s_n)$. Show that $\liminf s_n \le \liminf \sigma_n$.

Since $X$ and $Y$ are metric spaces, it suffices to show that if $\langle x_n:n\in\Bbb N\rangle$ is a convergent sequence in $X$ with limit $x$, then $\langle f(x_n):n\in\Bbb N\rangle$ is a convergent sequence in $Y$ with limit $f(x)$; in words, *f preserves convergent sequences*.

Suppose that $\langle x_n:n\in\Bbb N\rangle$ converges to $x$ in $X$. If there is an $n_0\in\Bbb N$ such that $x_n=x$ for all $n\ge n_0$, it’s trivially true that $\langle f(x_n):n\in\Bbb N\rangle\to f(x)$, so assume (by passing to a subsequence if necessary) that $\langle x_n:n\in\Bbb N\rangle$ is a sequence of distinct points. (Since $\langle x_n:n\in\Bbb N\rangle$ converges to $x$ and is not eventually constant at $x$, it cannot have a constant infinite subsequence: for each $n\in\Bbb N$ there must be an $m>n$ such that $x_k\ne x_n$ whenever $k\ge m$.)

For each $n\in\Bbb N$ set $K_n=\{x\}\cup\{x_k:k\ge n\}$; each $K_n$ is compact and infinite. (Why?) By hypothesis, therefore, each $f[K_n]$ is compact.

For convenience let $y=f(x)$, and let $y_n=f(x_n)$ and $H_n=f[K_n]$ for $n\in\Bbb N$. By hypothesis each $H_n$ is compact and infinite, so each contains a limit point. Fix $n\in\Bbb N$. For each $k\ge n$, $Y\setminus H_{k+1}$ is an open nbhd of $y_k$ that contains only finitely many points of $H_n$ (why?), so $y_k$ can’t be a limit point of $H_n$. Thus, for each $n\in\Bbb N$ the only possible limit point of $H_n$ is $y$ itself. From here you should be able to prove without too much trouble that $\langle y_n:n\in\Bbb N\rangle\to y$ and hence that $f$ is continuous.

- How many positive integer solutions are there to the equality $x_1+x_2+…+x_r= n$?
- How can I calculate non-integer exponents?
- Why an eigenvector exists iff $\operatorname{det} (A-\lambda I) = 0$?
- Logarithm proof problem: $a^{\log_b c} = c^{\log_b a}$
- If $f:[0,\infty)\to [0,\infty)$ and $f(x+y)=f(x)+f(y)$ then prove that $f(x)=ax$
- Geometrical interpretation of simplices
- $\displaystyle\sum_{k=0}^n \frac{\cos(k x)}{\cos^kx} = ?$
- Continuity from below and above
- An Integral Involving Brownian Motion
- Models of real numbers combined with Peano axioms
- Confusion about Homotopy Type Theory terminology
- Everywhere Super Dense Subset of $\mathbb{R}$
- If $a_{ij}=\max(i,j)$, calculate the determinant of $A$
- Binomial Coefficients with fractions
- If $\lim_{h\to 0} \frac{f(x_0 + h) – f(x_0 – h)}{2h} = f'(x_0)$ exists, is f differentiable at $x_0$?