# Tag Info

### Fastest way showing the limit exists without finding the limit?

Write your formulas in a matrix form: $$\begin{bmatrix}a_n\\b_n\end{bmatrix} = \begin{bmatrix}1&1\\r&1\end{bmatrix}\begin{bmatrix}a_{n-1}\\b_{n-1}\end{bmatrix}$$ i.e. $$w_n=Aw_{n-1}$$ ...
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### Does there exist $x$, $y \in \mathbb{N}$ such that $x^2 − y^2 = 19$

Notice that $19$ is prime. because of this, one of $x-y$ and $x+y$ must be $1$, and the other $19$. Therefore, if we solve the system $x-y = 1, x+y=19$, we can find a solution to this problem. You ...
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Accepted

### Does proper contraction on Hilbert space necessarily lead to convergence in norm to zero?

Let $\{a_n\}$ be a sequence of positive real numbers that increases to $1$, with the property that the sequence of products $$a_1,\ a_1a_2,\ a_1a_2a_3,\ a_1a_2a_3a_4,\ \ldots$$ converges to a ...
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### Prove that $f(x)=x^2$ is a contraction on each interval in $[0,0.5]$

Remark that $$|f(x)-f(y)| = |(x+y)(x-y)| \leq (|x|+|y|)|x-y| < 2a |x-y|$$ if $x$, $y \in [0,a]$. Now $2a<1$ by assumption.
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### an example of a complete space $X$ and a such mapping $T$ without fixed points & Show that if $X$ is compact then such $T$ has a unique fixed point.

Hint for (a): Apply the Mean Value Theorem to that function you were given. Hint for (b): Since $d(T(x),T(T(x))) < d(x, T(x))$ if $x\ne T(x)$, you must have $\inf d(x,T(x))=0$. (You should ...
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### Using connectedness to prove surjectivity...

Let $(x,y)\in\mathbb R^2$ and let $$g: u \mapsto x - f\left(y-f(u)\right)$$ \begin{align} \left|g(u) - g(v)\right|& = \left|x-f(y-f(u)) - x + f(y-f(v))\right|\\ &= \left|f(y-f(u))-f(y-f(v))\...
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### Fastest way showing the limit exists without finding the limit?

Again, not sure if this satisfies your criteria, but if we let $f : [0,\infty) \to \mathbb{R}$ by $$f(z) = \frac{r+z}{1+z}$$ then $f$ is decreasing, $0\leq f \leq r$, and $c_{n+1} = f(c_n)$. So if ...
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### prove there is no integer solution to x^2 -6 = 0 by contradiction

Assume an integer solution $x$ such that $x^2 = 6$ exists. An integer is either odd or even. The square of odd $x$ is odd (whereas $6$ is even), so $x$ must be even. Let $x = 2k$, where $k$ is an ...
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### Continuous function from set to proper subset is a contraction

Let $f:[0,1]\to [0,1)$ be given by $f(x)=\frac12\sqrt{x}$. Then the ratio $|f(x)-f(0)|/|x-0|$ grows without bound as $x\to0$.
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### Contraction, doubt on definition

Here they call it a 'contractive mapping', which I think makes no difference with the word 'contraction'. Well, you do not necessarily have fixed points.
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### Solve differential equation via successive approximations (contraction mapping principle)

Contractractivity on a rectangular domain To apply the fixed-point theorem one needs to fix a domain for the ODE and determine the Lipschitz constant there. In the most simple case the domain is ...
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You should have gotten a plot like this which gives a visible idea of convergence for $|t|\le 0.5$. Let's consider the interval $|x-1|<1$ in state space in the proof construction of the Picard ...