Proving $|a+b|\le |a|+|b|$ from $-|a|\le a \le |a|$

Adding

$$-|a|\le a\le |a| $$ and

$$-|b|\le b\le |b|.$$

you get

$$-(|a|+|b|)\le a+b\le|a|+ |b|,$$

which is

$$-(a+b)\le|a|+|b|\land a+b\le|a|+|b|$$ or $$|a+b|\le|a|+|b|.$$

Tags:

Inequality

Absolute Value

Alternative Proof

Algebra Precalculus

Related

If positive integers $a$, $b$, $c$ satisfy $\frac1{a^2}+\frac1{b^2}=\frac1{c^2}$, then the sum of all values of $a\leq 100$ is ... Why is this integral bounded? How do you find appropriate trig substitution for $\int \frac{\sqrt{16x^2 - 9}}{x} \, dx$? Can we represent an improper integral as $\int_{-\infty}^{\infty} f(x)\,dx = \lim_{a \to \infty} \int_{-a}^a f(x)\,dx$? Definition of topological space & open sets How can i find ${I_{n}=\int_{0}^{1}\frac {x^{2n}\ln x}{{(1-x^2)}{(1+x^4)^n}}dx{,n} \in N}$ A Hamiltonian inequality using uniform convexity and Taylor's formula (Evans PDE, §3.3.3, Lemma 4, (36)) How do concepts such as limits work in probability theory, as opposed to calculus? Closed form of $\int_0^\infty \arctan^2 \left (\frac{2x}{1 + x^2} \right ) \, dx$ Prove $(a^2+b^2+c^2)^3 \geqq 9(a^3+b^3+c^3)$ How many paths are possible such that the area of the square cut off is exactly half the area of the entire square? How Can I prove $\sum_{n=1}^{\infty}{\frac{n^4}{5^n}}=\frac{285}{128}$

Recent Posts