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Jun
17
comment countable open cover is reducible to a finite cover
What is your definition of countably compact and what have you tried?
Jun
11
comment Prove that if $b \mid c$ then $ab \mid c$?
$5\cdot 5=4$ ${}{}{}$
Apr
27
awarded  Informed
Jan
26
comment Geometric proof of complex number equation
$e^{i\theta}$ lies in the unit circle, $|z-w|$ is the distance between $z$ and $w$, and $\sqrt{2}$ can be seen as the length of the diagonal of a square of side 1.
Jan
16
awarded  Yearling
Jan
14
comment Extremely hard and stimulating (undergraduate) real analysis $problems$
Problems in Real Analysis: Advanced Calculus on the Real Axis, maybe?
Jan
3
comment Deceptively simple divisibility problem
@tmrlvi but there is no $k$ such that $9=2a+b\mid (a+b)^k=7^k$.
Jan
2
comment How can I prove these generalizations of Weierstrass?
Glad it helped.
Jan
2
comment How can I prove these generalizations of Weierstrass?
Okay. $\lim_{x\to +\infty}f=+\infty$ iff for every real number $M$ there is a real number $b$ such that $f(x)>M$ if $x>N$ (this is the definition). Now, taking $M=f(a)$, we find a real number $b$ ($>a$ because the domain of $f$ is $[a,+\infty)$) such that $f(x)>f(a)$ if $x>b$. By the extreme value theorem (Weierstrass' th.), there is a real number $c\in [a,b]$ such that $f(c)\le f(x)$ for all $x\in [a,b]$. In particular, $f(c)\le f(a)$ and $f(a)<f(x)$ for $x\in (b,+\infty)$, so $f(c)\le f(x)$ for all $x\in [a,+\infty)$. Hence $c$ is a minimum of $f$.
Jan
2
comment How can I prove these generalizations of Weierstrass?
Which part do you find problematic to formalize?
Jan
2
comment How can I prove these generalizations of Weierstrass?
You can reduce the statements to the ones you know. For example, in $(1)$, by definition of $\lim_{x\to +\infty} f=+\infty$, there is some real number $b$ such that $f(x)>f(a)$ if $x>b$. Now you can find the minimum of $f$ in $[a,b]$.
Dec
20
comment elementary topology exercises reference
See Elementary Topology: Problem textbook
Dec
20
awarded  Constituent
Dec
16
answered Irreducible in $\Bbb Q[x]$
Dec
15
awarded  Custodian
Dec
15
reviewed No Action Needed Are there number systems corresponding to higher cardinalities than the real numbers?
Dec
10
comment The number of one-dimensional vector spaces in a field
possible duplicate of How to count number of bases and subspaces of a given dimension in a vector space over a finite field?
Dec
10
comment If $\sum x^6_n$ converges, then $\sum x^7_n$ converges too
Hint: In $[0,1]$, we have $x^7\le x^6$.
Dec
9
awarded  Caucus
Dec
8
comment Prove or disprove: there is an integer $x$ so that $x \equiv 2$ (mod 6) and $x \equiv 3$ (mod 9).
Glad it helped you, @GabrielH