Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

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How to show that a Cauchy sequence of sequences converges

Let $c_0$ be the space of real-valued sequences $\{x_n\}$ which converge to zero, equipped with the metric $d(\{x_n\}, \{y_n\}) = sup_n |x_n − y_n|$. I want to show that the metric space $(c_0, d)$ is ...
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Finding a special subsequence of any Cauchy sequence

Let $(X,d)$ be a metric space and let $(x_n)$ be a Cauchy sequence in $X$. Let $(\epsilon_n)$ be a sequence of real numbers and decrease to $0$. Show that there is a subsequence $(x_{n_k})$ of $(x_n)$ ...
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Is this space a Hilbert Space?

I have a space of continuously differentiable functions on [a, b] with the dot product defined in this way: $x \cdot y = \int_a^b \! [x(t)y(t) + x'(t)y'(t)] \, \mathrm{d}t.$ Is this space a Hilbert ...
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Using the Uniform Cauchy Criterion theorem

Suppose $(f_n)$ is defined, continuous on $[a,b]$, and differentiable on this open interval. Then $c \in [a,b]$ and $(f_n(c))$ converge $(f'_n)$ and converge uniformly on $(a,b)$ respectively. How ...
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Proving convergent sequences are Cauchy sequences

Prove that if $x_n \rightarrow a, n \rightarrow \infty$ then $\{x_n\}$ is a Cauchy sequence. I believe I have found the proof as follows, wondering if there are any simpler methods or added ...
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Prove the space of bounded sequences is Banach

http://www.math.ucla.edu/~tao/resource/general/121.1.00s/exam1sol.pdf Here is a proof, but I cannot fully understand why it does not give a proof that $x$ is a bounded sequence (i.e. $x$ is in the ...
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Baby Rudin Exercise 4.13 Alternate Proof Verification

I would like to know if my proof of ex 4.13 is correct. Thanks! Exercise 4.13 in Rudin asks: Let $E$ be a dense subset of a metric space $X$, and let $f$ be a uniformly continuous real function ...
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Is $x^n$ Cauchy in $(C[0, 1], \|\cdot\|_{\infty})$?

Consider the sequence of functions $$f_n(x) = x^n, \quad x \in [0, 1].$$ Is this sequence Cauchy in $(C[0, 1], \|\cdot\|_{\infty})$? The pointwise limit is not ...
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Find a limit of the recursive sequence

The task is to prove sequence convergence and find a limit. $x_0=0$ $x_1=1$ $x_{n+1}=\frac {x_n + n \cdot x_{n-1}} {n+1}$ I have computed some values of a sequence to build up some idea of the data: ...
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How to prove that if $\sum _{n=1}^{\infty }a_n\:$ converges then $\sum _{n=1}^{\infty }a_na_{2n}\:$ converges?

How to prove that if $\sum _{n=1}^{\infty }a_n\:$ converges then $\sum _{n=1}^{\infty }a_na_{2n}\:$ converges? Note: $a_n \in \mathbb R$ I tried to prove it using cauchy criterion The idea was to ...
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To show that a recursively defined sequence by $x_1=\frac12$ and $x_{n+1}=\frac{x_n^3 + 2}{7}$ is Cauchy - How?

Show that the sequence defined by $x_1$ = $\frac{1}{2}$ and $x_{n+1} = \frac{x_n^3 + 2}{7}$ for $n \in N$ satisfies the Cauchy criterion. I don't understand how to go about this problem. Is it ...
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