Prove inequality $(1+\frac{1}{2n})^n<2$ How can I prove inequality
$$
(1+\frac{1}{2n})^n<2
$$
for $n\in\mathbb{N}$ by using induction proof method?
I have tried to write that:
for $n=1$ it's true. So let's suppose that $(1+\frac{1}{2n})^n<2$. and now I don't know what do nect because I know that $(1+\frac{1}{2(n+1)})^{n+1} > (1+\frac{1}{2n})^n$ but I don't know how to show that $(1+\frac{1}{2(n+1)})^{n+1} < 2$.
In this task I cannot use the definition of $e$ number.

It is enough to show that $\log(2)\geq \frac{1}{2}$, since
$$ 2^{1/n} = e^{\frac{\log 2}{n}} > 1+\frac{\log 2}{n} \stackrel{\color{red}{?}}{\geq} 1+\frac{1}{2n}.$$
is granted by the convexity of $e^x$. On the other hand $\log(2)>\frac{1}{2}$ is equivalent to $e<4$, and
$$ e = \sum_{k\geq 0}\frac{1}{k!}=\frac{8}{3}+\sum_{k\geq 4}\frac{1}{k!}<\frac{8}{3}+\frac{1}{6}\sum_{k\geq 4}\frac{1}{4^{k-3}}=\frac{49}{18}. $$
As an alternative,
$$ 0 < \int_{0}^{1}x^3(1-x^3)e^{-x}\,dx = \frac{1158}{e}-426 $$
directly gives $e<\frac{193}{71}$. Or, more elementary:
$$ \log(2)-\frac{1}{2}=\int_{0}^{1}\frac{dx}{1+x}-\int_{0}^{1}\frac{dx}{1+1}=\frac{1}{2}\int_{0}^{1}\frac{1-x}{1+x}\,dx > 0.$$

Yet another approach: for any $n\geq 1$, $2^{1/n}$ is the geometric mean of $\frac{n+1}{n},\frac{n+2}{n+1},\ldots,\frac{2n}{2n-1}$. By the AM-GM inequality it follows that
$$\begin{eqnarray*} 2^{1/n} < \frac{1}{n}\sum_{k=0}^{n-1}\left(1+\frac{1}{n+k}\right)&=&1+\frac{1}{n^2}+\frac{1}{n}\sum_{k=1}^{n-1}\frac{1}{k+n}\\&<&1+\frac{1}{n^2}+\frac{1}{n}\sum_{k=1}^{n-1}\frac{1}{\sqrt{k+n}\sqrt{k+n-1}}\end{eqnarray*} $$
and by the Cauchy-Schwarz inequality (see also page 8 of my notes) we have
$$ \sum_{k=1}^{n-1}\frac{1}{\sqrt{k+n}\sqrt{k+n-1}}\leq \sqrt{\sum_{k=1}^{n-1}1\sum_{k=1}^{n-1}\left(\frac{1}{k+n-1}-\frac{1}{k+n}\right)}=\frac{n-1}{\sqrt{n(2n-1)}},$$
so $2^{1/n}< 1+ \frac{1}{n\sqrt{2}}+\frac{1}{n^2}$ and actually $2^x< 1+\frac{x}{\sqrt{2}}+x^2$ for any $x$ in a pointed neighbourhood of the origin. By replacing $x$ with $-x$ and reciprocating we get
$$ 2^x > \frac{1}{1-\frac{x}{\sqrt{2}}+x^2} $$
and
$$ 2^{1/n} > \frac{1}{1-\frac{1}{n\sqrt{2}}+\frac{1}{n^2}} $$
still is stronger than needed.
