Tagged Questions

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Showing $a_n=\sin(n)$ does not converge

Show that $a_n=\sin(n)$ does not converge My idea: Take two subsequences: $a_{n_k}=\sin(\frac {\pi k} 2)$ , $a_{n_l}=\sin(\frac {2\pi l} 3)$ So: $\forall n$ : $\lim_{n\to\infty} a_{n_k}=1$, ...
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Deducing the rules for inequalities from the order properties.

I'm trying to relearn calculus but going more in-depth this time, meaning doing all exercises and especially the ones dealing with proofs. I'm using the Thomas calculus textbook. Thomas starts by ...
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Using epsilon -delta definition to show that $\lim_{x \rightarrow 2} (x^3 + \sqrt[3]{x}) = 8+ \sqrt[3]{2}$

I'm trying to use the $\varepsilon$-$\delta$ definition to show that $\lim\limits_{x \rightarrow 2} (x^3 + \sqrt[3]{x}) = 8+ \sqrt[3]{2}$. I already know how to prove continuity for cubic root using ...
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Non-negative, continuous function with integral [duplicate]

Let there be an integrable, non-negative function $f$ in a range $[a,b]$. If the integral $\int_a^b f(x) \, dx$ equals $0$, prove that $f(x)=0$ for every $x$ for which $f$ is continuous. I have ...
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Trouble understanding math proofs

*edit Even though there are already answers to my question, I appreciate anyone that offers their advice! I am not sure if this is the right place to ask this but I usually ask for help here. I am a ...
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Proof that the limit of a sequence is equal to the limit of its partial sums divided by n

Let $\{ x_n \}_n$ be a sequence of real numbers. Suppose $\lim_{n \to \infty}x_n=a.$ Show that $$\lim_{n \to \infty} \frac{x_1+x_2+...+x_n}{n}=a$$ As it is my first proof I'm not really sure ...
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alternate proof of theorem 1.21 of Baby Rudin

I wanted to ask if anyone has tried to prove theorem 1.21 from Baby Rudin book (on existence and uniqueness of positive real n-th root of a positive real number) differently and would care to check my ...
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Review of solution: Prove $\liminf({a_n}) \ge \liminf({b_n})$

${a_n} \ge {b_n}\forall n \in$ Prove: $\liminf({a_n}) \ge \liminf({b_n})$ I proved it by contradiction. Let's assume $\liminf({a_n}) < \liminf({b_n})$. $a := \liminf({a_n})$ $b := \liminf({b_n})$ ...
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Proof that the derivative of a linear function is $0$.

We have a function $f(x)=x$ which is defined and is continuous on the set $S$ of all real numbers. The derivative at point $x$, $$f'(x)=\lim_{h\to 0} \frac{f(x+h)-f(x)}h$$ Using the theorems of ...
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Proving the derivative is $0$ at the extremum and all derivatives are $0$.

The pictures below show the proof that Apostol uses in his book. I can't understand why Apostol introduces the function $Q(x)$ and proves the theorem by contradiction using the sign preserving ...
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Proving that $\sin(x)$ is continuous at $0$

Given: $|\sin x| < |x|$, valid for $0<|x|<\frac12\pi$ (EDIT: $\frac12 \pi$ not $\frac1{2\pi}$) Conclusion: $\lim_{x\to 0}\sin(x) = 0$, also expressed as $|\sin(x)| < \epsilon$ if ...
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Proving that $1/x$ and $1/x^2$ limit does not exist

1) If I am to prove that limit of $\frac1x$ doesn't exist at $x\to0$ is it sufficient and rigorous enough to show that the left hand and the right hand limits are not equal(EDIT: are not equal ...
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Requesting feedback on proof of theorem

I'm trying to self study my way through Apostol's calculus and have just started. Having completed an undergraduate degree in physics some time ago, the math courses I took mostly focused on ...
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Using inequalities and limits

Is it possible to say: $$If \ f(x) \ and \ g(x) \ both \ have \ limits \ as \ x\to p\ and \ f(x) \le g(x), \ then \lim_{x \to p} f(x)\le \lim_{x \to p} g(x).$$ My proof(Edit: Proof is wrong due to ...
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Proving the limits of the sum of two functions is equal to the sum of the limits

I am new to proving in math so I want to know if this informal proof of limits is possible: Theorem: If $\lim_{x \to a}f(x)=A$ and $\lim_{x \to a}g(x) = B$, then $$\lim_{x \to a}[f(x)+g(x)]=A+B$$ ...
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Prove that if $n \cdot 2^{-t} <0.01$ then $n \cdot 2^{-t} <\frac{1}{101}$

Is the following theorem true? If $n \cdot 2^{-t} <0.01$ then $n \cdot 2^{-t} <\frac{1}{101}$ for $t,n \in \mathbb{N}$. I've tried basic induction but that has led me nowhere, same with ...