1.2 Inequalities

Inequalities come up everywhere in maths! For example, they can be used in statistics for estimation (Markov/Chebyshev inequalities), they can be used as constraints in optimisation problems (see Section 3.1 of this Wikipedia link.), and quite famously appear in Quantum Mechanics. In this latter case, we have the Heisenberg Uncertainty Principle, and this inequality states that you can’t simultaneously know the position and momentum of a quantum particle, such as an electron.

Most of the inequalities in this course will be based on the absolute value, which is defined as follows:
Definition 1.3: (Absolute Value)
For \(x \in \mathbb{R}\), the absolute value of \(x\) is given by \[\begin{align*} \lvert x \rvert = \begin{cases} x \quad &\text{if} \; x \geq 0,\\ -x \quad &\text{if} \; x < 0 \end{cases}\;\; = \max\lbrace x, -x \rbrace. \end{align*}\]
The absolute value has the following properties:
Proposition 1.2:

For \(x,y \in \mathbb{R}\): \[\begin{gather*} x \leq \lvert x \rvert,\quad -x \leq \lvert x \rvert,\quad \lvert -x \rvert = \lvert x \rvert\quad \text{and}\quad \lvert x y \rvert = \lvert x \rvert \lvert y \rvert. \end{gather*}\]

Now we come on to what I consider to be the most important thing in this course.

Theorem 1.3: (Triangle Inequalities)

For \(x,y\in\mathbb{R}\):

  • \(\lvert x + y \rvert \leq \lvert x \rvert + \lvert y \rvert\), and
  • \(\left\lvert \lvert x \rvert - \lvert y \rvert \right\rvert \leq \lvert x - y \rvert.\)

The first of these is known as the Triangle Inequality, and the second is the Reverse Triangle Inequality. Why do I think this is so important? This will come up in almost any course you take at university that uses analysis! If you’re studying vector calculus, fluid mechanics, statistics, probability, or anything that’s not abstract algebra, there’s guaranteed to be a proof or technique which involves an inequality of this form! So if you only learn one result from Analysis 1A, make it this one. Finally, there’s one more inequality to mention — the binomial inequality.
Proposition 1.4: (Binomial Inequality)

We have \(\forall n \in \mathbb{N}_0\) (i.e. all the natural numbers with \(0\)), and \(\forall x \geq -1\), \[(1 + x)^n \geq 1 + nx.\]