Can anyone help me understand where I am misunderstanding the principles of induction? I am trying to understand induction by using it to prove a statement is false (below) but I am not "getting it".

Here is the principle of Mathematical Induction (from http://www.math.toronto.edu/oz/turgor/Induction.pdf) and below it is my contradicting rationale:

Here are my thoughts going through this step by step. Please help point out where my logic is flawed:Principle of Mathematical Induction

If it is known that

(1) some statement is true for n = 1

(2) assumption that statement is true for n implies that

(n + 1)

then the statement is true for all positive integers

Thank you so muchstep 1

create a statement that is true for n=1:

ok here is my statement that I created to test this out:

for all non-negative integers, 1+2+3+...n =< 2n

for n=1, this statement holds true

step 2

assume that if statement is true for n, then it is true for n+1

ok so the statement is also true for the new "n", which is 2. supposedly this statement should be true for all positive integers based on the above principle of mathematical induction

contradiction

what if n=4? the statement is now false. but according to my understanding of induction, it should be true for any positive integer...