How to prove every row-echelon matrix is upper triangular In the Nicholson's linear algebra book, I encountered this statement:

Every row-echelon matrix is upper triangular

I tried to prove that using the three conditions for row-echelon form but didn't get result.
This is the definition of the row-echelon form:


Assume that a matrix $A$ is in row echelon form. Let $r$ be the number of nonzero rows in $A$. Let $a_{ij_{i}}$ denote the leading $1$ in row $i$.
Notice that $1\leq j_{1} < j_{2} <\ ...<\ j_{r-1}< j_{r} \ $and so $n\leq j_n$ for $n=1,2,...,r$.
Also, for rows $i\leq r,\ $ $a_{ik} = 0$ for $k<j_{i} \implies a_{ik} = 0 \ $ for $k < i\ $ since $i\leq j_i$.
Lastly, rows $i>r$ are zero rows, so $a_{ik} = 0$ for $k<i$.
$\therefore A$ is an upper triangular matrix. 
