# How to prove an inequality by mathematical induction? [on hold]

How to prove the following inequality by mathematical induction?

$$x + \frac 1x \ge 2, x \gt 0$$

I am aware of this.

First, I have to prove $$P(1)$$; then $$P(n+1)$$.

I am stuck at $$P(n+1)$$ because I do not know how to add the plus one to it. thanks.

EDIT 4/10/19

I misunderstood the problem; I thought I needed to prove the above inequality by mathematical induction; I learnt that I just need to prove it. I think the easiest way to prove this inequality is by constructing a direct proof.

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• And what is $P(n)$? – José Carlos Santos Mar 19 at 23:09
• Induction on what? – Bernard Mar 19 at 23:10
• Is $x$ here just positive integers, or all positive real numbers? – Minus One-Twelfth Mar 19 at 23:14

No way! Induction is only for integers.

By the way here’s another way to solve it:

$$(\sqrt{x}-\frac{1}{\sqrt{x}})^2\ge 0$$ $$\Rightarrow x +\frac{1}{x}-2\ge 0\Rightarrow x+\frac{1}{x}\ge 2$$

If the inequality is supposed to be proved for positive integers $$x$$ then you get $$P(n+1)$$ from $$P(n)$$ as follows: $$n+1+\frac 1 {n+1}=n+\frac 1 n +1+\frac 1 {n+1}-\frac 1 n \geq 2+1-\frac 1 {n(n+1)}\geq 2$$ because $$n(n+1) \geq 1$$ and $$\frac 1 {n(n+1)} \leq 1$$

• what if $n=1$?? then $1(1+1) \geq 1$ may not be true – James Mar 20 at 0:06
• @JimmySabater I don't see any problem here when $n=1$. – Kavi Rama Murthy Mar 20 at 0:10
• What do you even mean by saying $2 \geq 1$ may not be true? – Kavi Rama Murthy 2 days ago

I think the easiest way to prove this inequality is by direct proof where you simplify the inequality, thereby obtaining a true statement. Then you try to check the algebra is still valid; like so:

$$x + \frac 1x \ge 2$$

$$x(x + \frac 1x) \ge 2x$$

$$x^2 + 1 \ge 2x$$

$$x^2 - 2x + 1 \ge 0$$

$$(x - 1)^2 \ge 0$$

Once you simplify, you prove it by reversing the process and checking the algebra is correct; like so:

$$x^2 - 2x + 1 \ge 0$$ $$x^2 + 1 \ge 2x$$ $$x + \frac 1x \ge 2$$

The result follows if one puts $$a=x$$ and $$b=1/x$$ on the famous AM-GM inequality

• The question clearly says 'by mathematical induction'. – Kavi Rama Murthy Mar 20 at 0:10