Originally Posted by

**metlx** Hi,

I'm stuck on the following problem:

Let $\displaystyle h \ \epsilon \ [0,1]$

Show that for every $\displaystyle n \ \epsilon \ N$

$\displaystyle (1 + h)^n \leq 1 + (2^n -1)h$

is true..

I tried with math induction:

$\displaystyle (1) \ \ \ n = 1$

$\displaystyle 1 + h \leq 1 + (2 - 1)h$

$\displaystyle 1 + h \leq 1 + h \ \ \ \surd $

$\displaystyle (2) \ \ \ n \Rightarrow n + 1$

$\displaystyle (1 + h)^{h+1} \leq 1 + (2^{n+1} - 1)h$

$\displaystyle (1+h) (1+h)^n \leq 1 + (2 \cdot 2^n - 1)h$

$\displaystyle (1+h) (1+h)^n \leq 1 + 2h \cdot 2^n - h$

$\displaystyle (1+h) (1+h)^n \leq 1 - h + 2(h \cdot 2^{n-1})$

now i'm lost. don't know how to continue.

By the way, how do I write the "is element of" symbol? did i use the correct one above?

and is there any way i could skip the [tex][/ math] in every line?