In many inequality problems, we often invoke a concept of "smoothing principle", where we have to move two elements closer together and repeating that until achieving a certain relation ... calculus inequality

2016年4月16日 · By algebraic inequalities I mean inequalities like Cauchy's inequality, the AM-GM inequality etc. I need it for the International Mathematics Olympiad (IMO), so I hope I can find some books that introduce inequalities on that level.

2019年2月9日 · As an olympiad-participant, I've had to solve numerous inequalities; some easy ones and some very difficult ones. Inequalities might appear in every Olympiad discipline (Number theory, Algebra, Geo...

Dividing an inequality by a positive number retains the same inequality. But, multiplying by $-1$ is the same as switching the signs of the numbers on both sides of the inequality, which reverses the inequality: $$ \tag{1} a\lt b\quad\iff -a\gt -b. $$ You should be able to convince yourself why the above is true by looking at the number line ...

That's actually the definition of the concept of "strictly decreasing function". So the fact that when you multiply by a negative number, you invert the inequality relation, is the same as saying that multiplication by a negative number is a strictly decreasing function.

2015年9月13日 · Your first example is definitely correct. The chained inequality $5 < x < 10$ formally has the same meaning as 5 < x & x < 10, that is, "5 is less than x and x is less than 10", equivalent to "x is greater than 5 and less than 10", or in interval notation $(5, 10)$.

I'm not sure what you expect exactly, but here is how I would do the inequality you mention. We start with the base step (as it is usually called); the important point is that induction is a process where you show that if some property holds for a number, it holds for the next.

2016年10月18日 · In the following completed proof, in the highlighted steps, it seems as though the proof is defending its claim by showing that $2(\\sqrt{k+2}-1) &lt; (\\text{the lower highlighted part in the proof}...

2013年2月19日 · for CW substitute vector = x-ty, for triangle inequality vector = x+y for CW, after dotting x -t y with itself let t = ( x . y )/( y . y ), for triangle ineq. after dotting x + y with itself and getting a quadratic equation with a dot product in the middle, use CW to show that this quadratic is less than or equal to the same quadratic with the ...

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