Seems to be true but I want to make sure:
$n \leq 2^{n-1}, n > 0$
Seems to be true but I want to make sure:
$n \leq 2^{n-1}, n > 0$
For $n=1$ we get equality. For $n\ge 2$ we get by Bernoulli $$2^{n-1}=(1+1)^{n-1} \ge 1+(n-1)\cdot 1 = n.$$
Yes.
You add $1$ to one side and multiply the other by $2$. Wherever $x>1\implies x=1+\lambda (\text{s.t. }\lambda>0)$, we have:
$1+x=2+\lambda$
$2x=2+2\lambda$
with the latter clearly being greater.
Our first comparison is $1\leq 1$ then our second (the important one for my point), is $2\leq 2$
How about a simple induction?
The statement holds for $n=1$. Assume the statement holds for $n$, that is, $n \le 2^{n-1}$. Need to show that it holds for $n+1$, that is $n+1 \le 2^{n}$.
We have $n+1 \le 2^{n-1} + 1$ according to the assumption. Since $2^{n-1} + 1 \le 2^n$ for all $n \ge 1$ (as $2^n-2^{n-1}=2^{n-1} \ge 1)$, it is also true that $n+1 \le 2^n.$
As said before, this the inequality is true. In fact, you can graph the function of the left side and the right side and see that there is only one intersection at one for values greater than zero.
To see this more simply, take the logarithm of both sides. Note, this approach is valid because the ln is a one-to-one function. You have:
$$ln(n) \le (n-1)ln(2) = ln(2)n-ln(2)$$ $$ \Rightarrow ln(n) + ln(2)=ln(2n) \le ln(2)n$$ Proof:
It is a common fact in math that for any natural number $n \gt 0$, $\ ln(n) \le \sum_{i=1}^n \frac{1} n $ $$\Rightarrow ln(2n) \le \sum_{i=1}^n \frac{1} {2n}= \frac{1}{2}\sum_{i=1}^n \frac{1} n $$ Note: $\frac{1}{2} \le ln(2)$ $$\Rightarrow ln(2n) \le ln(2)\sum_{i=1}^n \frac{1} {n} \le ln(2)\sum_{i=1}^n 1 = ln(2)n $$ Therefore, the inequality is correct.
For $n=1$ we get equality. For $n \ge 2$, consider the list of $n-1$ positive numbers:
$$[\underbrace{1,1,\ldots,1}_{n-2},n]$$
By AM-GM inequality,
$$\begin{align*} \sqrt[n-1]{n\prod_{i=1}^{n-2}1} &\le \frac{n + \sum_{i=1}^{n-2}1}{n-1}\\ \sqrt[n-1]n &\le \frac{2n-2}{n-1}\\ n&\le 2^{n-1} \end{align*}$$