Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, and integration through the fundamental theorem of calculus.

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16
votes
4answers
434 views
+100

How do you prove that $\{ Ax \mid x \geq 0 \}$ is closed?

Let $A$ be a real $m \times n$ matrix. How do you prove that $\{ Ax \mid x \geq 0, x \in \mathbb R^n \}$ is closed (as in, contains all its limit points)? The inequality $x \geq 0$ is ...
5
votes
0answers
102 views
+100

Arithmetic implications of different ways to geometrically construct an Hilbert's curve

I have a question on the relation between the geometric and the arithmetic representation of the Hilbert's space-filling curve. Geometric representation: consider the Hilbert's curve $f_h:[0,1]\...
3
votes
1answer
84 views
+50

Drawing large rectangle under concave curve

Let $f$ be a continuous concave function on $[0,1]$ with $f(1)=0$ and $f(0)=1$. Does there exist a constant $k$ for which we can always draw a rectangle with area at least $k\cdot \int_0^1f(x)dx$, ...
1
vote
1answer
155 views
+100

Theorem regarding Change of Variables in finite dimesnion

My question is based on Change of Variables in Multiple Integrals II Peter D. Lax > It is not necessary to read the paper before answering this question.The author tried to prove change of variables ...
4
votes
1answer
179 views
+50

$f \in C^2(\mathbb R)$ , $(f(x))^2 \le 1$ ; $(f'(x))^2+(f''(x))^2 \le 1 $ ; then is $(f(x))^2+(f'(x))^2 \le 1 $?

Let $f \in C^2(\mathbb R)$ be such that $$(f(x))^2 \le 1 ; (f'(x))^2+(f''(x))^2 \le 1 , \forall x \in \mathbb R$$ Then is it true that $(f(x))^2+(f'(x))^2 \le 1 , \forall x \in \mathbb R$ ? I ...
1
vote
1answer
56 views
+150

Is it possible to find the partial sum?

Let, $a_n=\frac {3^{n+1}}{1+2^{n+1}},n\geq0.$ Let $S_n$ be the partial sum defined by $$S_n=\sum_{i=0}^{n}a_i.$$ Is it possible to write a closed formula for $S_n.$ I have no idea how to do this. Any ...