Intereting Posts

$H^{-1}(\Omega)$ given an inner product involving inverse Laplacian, explanation required
Prove $\sin(1/x)$ is discontinuous at 0 using epsilon delta definition of continuity
Sum of all the positive integers problem
$f$ not differentiable at $(0,0)$ but all directional derivatives exist
If $0\leq A\leq B$ on Hilbert space and $A^{-1}$ exists, show that $A^{-1}\geq B^{-1}$
Solution for the trigonometric-linear function
Does such a finitely additive function exist?
Minimum degree of a graph and existence of perfect matching
Why are vector spaces not isomorphic to their duals?
Proof that $e=\sum\limits_{k=0}^{+\infty}\frac{1}{k!}$
How to show the statement is false?
Expected number of rolling a pair of dice to generate all possible sums
Are there countably many infinities?
What Gauss *could* have meant?
On the Paris constant and $\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\dots}}}}$?

How can I show that the space of continuous real valued functions on R with compact

support in the usual sup norm metric is not complete ?

I know that this result can be proved by using the fact that the given space is dense in the space of all continuous functions that vanish at infinity which is complete ,

but I want a bit easier proof of this as I have not studied measure theory . I was

trying to solve it directly from the definition of completeness.

Thanks for any help.

- Proving the every subset of $M$ is clopen.
- Proofs for complete + totally bounded $\implies$ compact.
- Showing $(C, d_1)$ is not a complete metric space
- Are compact subsets of metric spaces closed and bounded?
- Let $X$ be a Moore space and $e(X)=\omega$. Is it metrizable?
- Sketch the open ball of a metric

- Prob. 3, Chap. 3 in Baby Rudin: If $s_1 = \sqrt{2}$, and $s_{n+1} = \sqrt{2 + \sqrt{s_n}}$, what is the limit of this sequence?
- Measuring $\pi$ with alternate distance metrics (p-norm).
- Understanding the definition of Cauchy sequence
- Continuous images of open sets are Borel?
- Distance of a matrix from the orthogonal group
- Metric Space Axioms
- Metrizability of a compact Hausdorff space whose diagonal is a zero set
- Is there a positive function $f$ on real line such that $f(x)f(y)\le|x-y|, \forall x\in \mathbb Q , \forall y \in \mathbb R \setminus \mathbb Q$?
- A topology on the set of lines?
- What about the continuity of these functions in the uniform topology?

Take a particular example of a continuous function that goes to $0$ at $\pm \infty$, and a sequence of continuous functions of compact support that converges uniformly to it. This is a Cauchy sequence …

- How many ordinals can we cram into $\mathbb{R}_+$, respecting order?
- Is the tensor product of non-commutative algebras a colimit?
- Cardinality of a vector space versus the cardinality of its basis
- Another version of the Poincaré Recurrence Theorem (Proof)
- Let $p$ be a prime integer. Show that for each $a ∈ GF(p)$ there exist elements $b$ and $c$ of $GF(p)$ satisfying $a = b^2 + c^2$.
- Examples of open problems solved through short proof
- Why are duals in a rigid/autonomous category unique up to unique isomorphism?
- Reduction from Hamiltonian cycle to Hamiltonian path
- On the general form of the family $\sum_{n=1}^\infty \frac{n^{k}}{e^{2n\pi}-1} $
- Why is it impossible to define multiplication in Presburger arithmetic?
- Prove the inequality $x+\frac{1}{x}\geq 2$?
- $M,N\in \Bbb R ^{n\times n}$, show that $e^{(M+N)} = e^{M}e^N$ given $MN=NM$
- $\nabla U=0 \implies U=\mathrm{constant}$ only if $U$ is defined on a connected set?
- Gromov-Hausdorff distance and the “set of all sets”
- Prove $\left| \int_a^b f(t) dt \right| \leq \int_a^b \left| f(t) \right| dt$