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"Nothing is more important than to see the sources of invention which, in my opinion, are more interesting than the inventions themselves."

--Gottfried von Leibniz (1646-1716)


6h
revised is there a pair of functions that meets the next requirement?
edited tags
23h
comment Intuition for high school students regarding square roots and logarithms
A better question to pose might be "why should that be right"?
2d
comment Using integration by parts, show that $\displaystyle \int e^{2x} \sin x dx=\dfrac{1}{5}e^{2x}(2\sin x-\cos x)+c$
No. You're just solving an equation of the form $I=B-4I$ for $I$.
2d
revised Writing roots of f(x) as f(a) for some a
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2d
revised What is the difference between the following sets.
edited tags
Sep
12
awarded  Nice Answer
Sep
12
awarded  Nice Answer
Sep
11
comment If $h_n\to h$ in the norm then $h_n\to h$ weakly
Are you working in a Hilbert space? If so, see, this. If not, you should tell what space you are working in.
Sep
10
comment Can there exist a continuous bijection from (0,1) to (0,1]? What about a continuous surjection?
For the first question, consider the behaviour of such a function, $f$, near the point $x$ where $f(x)=1$.
Sep
9
awarded  Nice Answer
Sep
9
awarded  Nice Question
Sep
9
answered Containment of $c_0$ or $\ell_p$
Sep
7
comment Nowhere Continuous Function
See this.
Sep
7
comment Sum of a geometric series
See this for ideas (the first answer, in particular).
Sep
5
comment $\lim_{n\to \infty} 1/n(1+1/2+…+1/n)$
You can estimate the sum using the Integral Test.
Sep
3
answered Prove that $\int_0^1f'(x)dx \leq f(1) - f(0)$.
Sep
3
comment Prove that $\int_0^1f'(x)dx \leq f(1) - f(0)$.
Let $f_n(x)={ f(x+1/n)-f(x)\over1/n}$. Note $(f_n)$ converges a.e. to $f'$. Note $\int_a^b f_n(x)=n\int_b^{b+1/n} f(x)\,dx -n\int_a^{a+1/n} f(x)\,dx$. Use Fatou.
Sep
3
comment Prove that the Cantor set cannot be expressed as the union of a countable collection of closed intervals
Yes, that's good advice. I didn't read the OP's first comment carefully... I meant to point out though, one does not have to regard the measure of the Cantor set, just the sum of the lengths of the sets comprising an $F_n$.
Sep
3
comment Prove that the Cantor set cannot be expressed as the union of a countable collection of closed intervals
You don't need any measure theory for what @VHP is suggesting. In the usual construction of the Cantor set, by removing "middle thirds", it is clear that for each $n$ the Cantor set is a subset of a finite union of pairwise disjoint intervals the sum of whose lengths is $(2/3)^n$ (from which it follows that the Cantor set has empty interior).
Sep
2
comment Show that $\lim a_n =0$ if $|\frac{a_{n+1}}{a_n}|<1, \forall n$
This isn't true. Perhaps you meant $\lim_{n\rightarrow\infty}|a_{n+1}/a_n|<1$?