# Tagged Questions

Questions on proving, manipulating and applying inequalities.

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### Inequality in the limit

Given that we have the following conditions: $f = O(\delta)$, $g = O(\delta^2)$, $f > 0, \delta > 0$, can we conclude that as $\delta \to 0^+$, $f+g>0$?
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### How would you prove inequality $2^n \gt n^{10}$ using induction

For the base case I can put a number such as $100$ for $n$ so $2^{100}\gt 100^{10}$. Ok so now the induction hyp: $2^{n+1} > (n+1)^{10}$ for $n \gt 101.$ where do I go from here? Also do I have ...
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### A counting problem on the integer lattice

Let $K$ be a subset of the integer lattice $\mathbb Z^2$such that it contains elements of the form $k=(k_1,k_2)$ where $k_1,k_2$ are integers and $k_2\neq 0$. Find $m$, an integer if possible, such ...
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### In $\triangle ABC$ show that $1 \lt \cos A + \cos B + \cos C \le \frac 32$

Here is what I did, tell me whether I did correct or not: \begin{align*} y &= \cos A + \cos B + \cos C\\ y &= \cos A + 2\cos\left(\frac{B+C}2\right)\cos\left(\frac {B-C}2\right)\\ y &= \...