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Apr
17
comment The sequence $\frac{2}{2-u_n}$ diverges
@Paul you need to be a little more careful than that. See my answer.
Apr
17
comment The sequence $\frac{2}{2-u_n}$ diverges
I suppose in proving 3) you could assume for contradiction that $u_n \to 2$, and then derive a contradiction. However, since you appear to be still learning proof style I would recommend avoiding proof by contradiction whenever direct proof is easy. I recommend this because if you assume something false for contradiction, then make a mistake in your proof and end up with an absurdity, you may be tricked into thinking you have succeeded.
Apr
17
revised The sequence $\frac{2}{2-u_n}$ diverges
added 57 characters in body
Apr
17
comment The sequence $\frac{2}{2-u_n}$ diverges
No no, steps 2) and 3) are completely independent of each other and should not refer to each other at all. You do not get to assume the sequence converges when you are trying to prove that it doesn't converge to $2$. You must show directly that it does not converge to $2$.
Apr
17
comment The sequence $\frac{2}{2-u_n}$ diverges
I'll boil down the argument. 1) Show the sequence $u_n$ is well defined. 2) Show that IF the sequence converged, THEN it must converge to $2$, and finally 3) Show that the sequence does not converge to $2$.
Apr
17
comment The sequence $\frac{2}{2-u_n}$ diverges
Where do you mean "here"? You don't need to assume the limit exists (this is what you are trying to prove is false.
Apr
17
reviewed Approve Joint density calculation
Apr
17
answered The sequence $\frac{2}{2-u_n}$ diverges
Apr
13
answered Finding equation of tangent to the curve perpendicular to the line
Apr
13
revised Finding equation of tangent to the curve perpendicular to the line
added 4 characters in body; edited tags
Apr
13
answered Why did Euler use e to represent complex numbers?
Apr
13
answered Is it possible to have $|f(x) - f(y)| \leq M\| x - y \|$ under such conditions?
Apr
13
comment If $f\in C[a,b]$ and if $\int\limits_a^bf(x)g(x)dx=0, $
You should choose $g \in C[a,b]$. You can construct such a $g$ piecewise like _/-----\_
Apr
13
answered If $f\in C[a,b]$ and if $\int\limits_a^bf(x)g(x)dx=0, $
Apr
6
revised Need some help on a non-example of equicontinuity
added 25 characters in body
Mar
21
awarded  Self-Learner
Mar
5
comment Does $P(X=a) = P(X=b) = \frac12$ maximize $\mathrm{Var}(X)$ over RVs a.s. taking values in $[a,b]$?
I now realize that the crux of my proof is your proof, and the rest of my proof is frivolous. (Well, I suppose knowing that symmetrizing increases variance is interesting...) So i'll accept your answer over my own.
Mar
5
accepted Does $P(X=a) = P(X=b) = \frac12$ maximize $\mathrm{Var}(X)$ over RVs a.s. taking values in $[a,b]$?
Mar
5
revised Does $P(X=a) = P(X=b) = \frac12$ maximize $\mathrm{Var}(X)$ over RVs a.s. taking values in $[a,b]$?
added 41 characters in body
Mar
5
answered Does $P(X=a) = P(X=b) = \frac12$ maximize $\mathrm{Var}(X)$ over RVs a.s. taking values in $[a,b]$?