Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [analysis]

Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

Mathematical analysis is the rigorous version of calculus. In fact, it investigates the theorems in calculus with enough care and deals with them more deeply, trying to generalize the ideas in calculus. You can consider a more specific tag instead: , , , , , , etc. For data analysis, use .

42884 questions
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Divergent sequence where $\lim_{n\to\infty}(a_{n+p}-a_n) = 0 $

I need to show that $(a_n)=\sin(\ln n)$ is a bounded sequence that has no limit, but for which $$\lim_{n\to\infty}(a_{n+p}-a_n )= 0\; \forall p \in \mathbb N.$$ I can show that the sequence is bounded and that it diverges, but I'm stuck with the…
selja
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Confusion about differentiability of a function between finite dimensional Banach spaces

I'm a little bit confused about something that should actually be simple. If we have a function f between finite dimensional Banach spaces. Then we have the implications: If f is partially differentiable with continuous partial derivatives, then f…
Amarus
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Proof that a strictly decreasing sequence of nested intervals boils down to a single point.

The nested intervals theorem says the following. If a sequence of intervals $\langle I_n\rangle$ is decreasing, then $\bigcap_{n=1}^{\infty} I_n$ is not empty. However, I'm trying to modify the theorem, say, if the sequence is strictly decreasing,…
Moreblue
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About the sequence $a_n=\{\pi^n\}$

Is the sequence $\{\pi^n\}=\pi^n-\lfloor\pi^n\rfloor$ dense? In other words for any given $\varepsilon>0$ and $t\in[0,1]$ is there a proper $n\in\mathbb{N}$ satisfying $|\{\pi^n\}-t|<\varepsilon$ ? (*) What is the condition on $q$ to make the…
K. Sadri
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derivative and cut off functions

Is there any way of constuct a cut off function which is $1$ in $B(0,\epsilon)$ and zero outside the ball $B(0,2\epsilon)$ and it's first and second derivative is smaller than $1/|x|^a$ , with $a<1/4$?
dizzy
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Existence of sequences converging to $\sup S$ and $\inf S$

If $S$ is a bounded set of real numbers, how can we prove that there are distinct sequences in $S$ that converge to $\sup S$ and $\inf S$? I'm not even sure how to begin on this problem. I know the set $S$ is bounded so it has a supremum and an…
devcoder
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Prove that $f$ is continuous in $\mathbb{R}$

Let $f:\mathbb{R} \to \mathbb{R}$ be a monotone function such that $f(V)$ is open for every open set $V\subset \mathbb{R}$. Prove that $f$ is continuous in $\mathbb{R}$. Any hint for proving this I will appreciate.
Thetexan
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What exactly is a derivative?

In calculus courses, we learn the classical derivative: $$f'(x)=\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$$ And the directional derivative: $$D_{\mathbf{v}}{f}(\mathbf{x}) = \lim_{h \rightarrow 0}{\frac{f(\mathbf{x} + h\mathbf{v}) -…
Red Banana
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Prove there is a unique $y:[0,1] \to \mathbb{R}$ solving $y(x) = e^x + \frac{y(x^2)}{2}$ for $x \in [0,1].$

The title is the problem statement, but to reiterate, Prove there is a unique $y:[0,1] \to \mathbb{R}$ solving $y(x) = e^x + \frac{y(x^2)}{2}$ for $x \in [0,1].$ Looking for hints/solutions, thanks in advance. Edit 6/10/16: Progress? If we define…
Merkh
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$\mathcal{H}$ is relatively compact iff every sequence in $\mathcal{H}$ has a convergent subsequence?

I'm trying to prove that Let $(Y,\rho_{Y}),(K,\rho_{K})$ a complete metric space and a compact metric space, respectively. Let, as well, $Z=\mathcal{C}^{0}(K,Y)$ the metric space of continuous fuctions, such that $K\longrightarrow Y$, with the…
elessartelkontar
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Showing that $\sup_{(x,y)}f(x,y)=\sup_x\sup_yf(x,y)=\sup_y\sup_xf(x,y)$

Can anyone help me prove this: Let $X$ and $Y$ be nonempty sets and $f:X\times Y\to\Bbb R$ such that $f(X\times Y)$ is bounded. Prove the following statement: $\sup_{(x,y)}f(x,y)=\sup_x\sup_yf(x,y)=\sup_y\sup_xf(x,y)\;.$ thank you
kamong
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Can $\mathbb R$ be partitioned into a countable number of dense subsets with same cardinality?

Is it possible to partition $\mathbb R$ into an countable number of disjoint dense subsets with the same cardinality? Furthermore, is it possible to partition the reals into an uncountable number of disjoint dense subsets with the same…
Patrick Tam
  • 443
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Prove that $\exists k\in \mathbb{N}^*$ such that $\|a-a_k\|<\varepsilon$

Let $E$ be a normed linear space, $C$ compact and $f:C\to C$ a function such that $\|f(x)-f(y)\|\geq \|x-y\|$ for all $x,y\in C$. Then $f$ is an isometry. Note: I'm having trouble trying to prove it! I feel stuck. My approach: Some hint says…
Valent
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Proof of Strong principle of Induction (T. Tao Analysis I)

I have no idea how to prove it by using only what the book has talked about so far. Can anyone help? The proof shouldn't be using set theory as set theory is only mentioned in the following chapter. The proof should only make use of the addition of…
Nick123
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Stuck at the proof of the existence of the partial fraction expansion

Let $P$ and $Q$ be complex polynomials with $deg(P) < deg(Q)$. Let $Q(z) = (z-z_1)^{k_1} (z-z_2)^{k_2}...(z-z_m)^{k_m}, z_i \in \mathbb{C} \text{ and } k_i \in \mathbb{N}$ be a complete decomposition. Then, I have to prove that there exists a…
Huy
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