TITLE: Why is mathematical induction a valid proof technique?
QUESTION [29 upvotes]: Context: I'm studying for my discrete mathematics exam and I keep running into this question that I've failed to solve. The question is as follows.

Problem: The main form for normal induction over natural numbers $n$ takes the following form:

$P(1)$ is true, and
for every $n, P(n-1)\to P(n).\qquad\qquad\qquad\text{[Sometimes written as $P(n)\to P(n+1)$]}$

If both 1 and 2 are true, then $P(n)$ is true for every $n$.
The question is to prove the correctness of the form above.

My work: My idea was to make a boolean statement and if it's a tautology in a true false table. That means the statement is always correct.
I've tried many times but I've failed.
Here is the original question in Hebrew:
n טענה כלשהי לגבי מםפר טבעי  $P(n)$ תהי 
אם מתקימיים שני התנאים הבאים:
1- הטענה $P(0)$ נכונה
2- לכל n>0, $P(n)$ נכונות הטענה $P(n-1)$ גוררת את נכונות הטענה 
אז $P(n)$ נכונה לכל מספר טבעי n
הוכיחו את נכונות משפט זה

REPLY [4 votes]: Formally, induction is an axiom or a consequence of other axioms.
Informally, it can be seen as similar to a proof by contradiction, perhaps something like: 

Let $n$ be the smallest value for which $P(n)$ is false.  
Unless
  $n$ is the smallest possible value in $\mathbb{N}$ (i.e. $1$, or if
  you prefer $0$), we know $P(n-1) \implies P(n)$, so if $P(n)$ is false
  then so too is $P(n-1)$ contradicting the definition of $n$. 
Meanwhile $n$ cannot be the smallest possible value in $\mathbb{N}$,
  as we know that $P(1)$ was true.