Direct Proof with example Can someone show what is the difference between direct proof and Mathematical induction.
I am Student of Master of information Security.And i have started studying number theory.My professor haven't expalined it that much.
Can someone help me please
 A: Let us take an example: show that $2^n> n$ for all $n\ge 1$. The direct proof is, for example, using the binomial formula
$$
2^n=(1+1)^n\ge 1+n >n.
$$
Induction would be, first the base case, $n=1$, which is true, and then
assuming that we know already $2^n>n$ to show the induction step that
$$
2^{n+1}>n+1.
$$
This follows from $2^{n+1}=2\cdot2^n>2n>n+1$ for $n>1$.
A: Proof using induction are usually natural numbers; it has a base case, and k to k+1 (or k+2 and others like it). So if the formula is right for 1, then 1+1 is also correct, then 2+1 is also correct... therefore the formula is proved.
The general proof uses a range of variety of methods, including looking for contradictions, infinite descent, and some others.
A: In direct proof, you start with some known and valid statements, and using other valid statements, you reach your target statement to be proven.  
While in mathematical induction, you first verify the statement to be proven for a small number say 0(or other value depending upon the problem) then, you assume that it is true for number $n$. Afterwards, using that very fact, you show that it is true for $n+1$ as well. Hence, this proves the statement. HOW?
you proved it true for 0. Then you proved if true for $n$, it is true for $n+1$ as well. So, true for 0 means true for 1, true for 1 means true for 2 and so on for all natural numbers.
