Changing an inequality, say I have $n$ < $n^{2}$ < $ 2^{n} $

Question

Say I have the statement
$n$ < $n^{2}$ < $ 2^{n} $

Would it then also be true that?

(I don't remember the word for this bear with me)

$2^{n}$ < $2^{n^{2}}$ < $ 2^{2^{n}} $

Answer 1

Your implication is $$ x < y \Rightarrow f(x) < f(y) $$ Such a function $f$ is called strictly monotonically increasing.

In your case $f(x) = 2^x$.

Answer 2

$$n< n^2<2^n \implies $$

$$n\ln(2)<n^2\ln(2)<2^n\ln(2) \implies $$

$$\ln(2^n)<\ln(2^{n^2})<\ln(2^{2^n})$$

take exponential which is strictly increasing at $\Bbb R$.

Answer 3

Recall that given $x_1>x_2$

• for $f(x)$ strictly increasing $\implies f(x_1)>f(x_2)$

• for $f(x)$ strictly decreasing $\implies f(x_1)<f(x_2)$

and given $x_1\ge x_2$

• for $f(x)$ increasing $\implies f(x_1)\ge f(x_2)$

• for $f(x)$ decreasing $\implies f(x_1)\le f(x_2)$