Show that two different representations to the base $k$ represent two different integers I would like to show:

Given two distinct, positive, integer representations in base $k$, say $\sum_{i=0}^na_ik^i$ and $\sum_{i=0}^mb_ik^i$ where $a_n \neq 0 \neq b_m$ and $a_i,b_i \in \{0,1,\ldots , k-1 \}$, prove that $$\sum_{i=0}^na_ik^i \neq \sum_{i=0}^mb_ik^i$$

I would also like to show this using the result that $$\sum_{i=0}^pc_ik^i \leq k^{p+1}-1$$ for every integer representation in base $k$. Additionally, I don't want to use the Basis Representation Theorem (that every basis representation is unique). 
What I have so far: I figured there are two cases to make $\sum_{i=0}^na_ik^i$ and $\sum_{i=0}^mb_ik^i$ be distinct. First, if WLOG $m>n$. Then we know $$\sum_{i=0}^na_ik^i\leq k^{n+1}-1 \leq k^m-1 < k^m \leq \sum_{i=0}^mb_ik^i$$ Then I moved on to the second case of $m=n$. For the two integer representations to be distinct then there must be some $i \in \{1,2,\ldots , n \}$ such that $a_i \neq b_i$. At this point I am stuck on how to show the two integer representations must be different numbers, using the result of $\sum_{i=0}^pc_ik^i \leq k^{p+1}-1$. Does anyone have an idea how to do this? Or to do away with cases?

This is a comment,
not an answer,
but the comment space is too small
to hold this.
I proved this result on
representation in general bases
over 40 years ago:
Let $\mathbb{B} =(B_j)_{j=0}^{\infty}$
be an increasing series
of positive integers
with $B_0 = 1$.
A positive integer $n$
is represented in
$\mathbb{B}$ if
$n$ can be written in the form
$n 
= \sum_{j=0}^{\infty} d_j B_j
$
where
$0 \le d_j < B_{j+1}/B_j
$.
(For the usual representation,
let $B_j = b^j$
for some integer $b \ge 2$.)
Then
(1) Every positive integer
can be represented in a form with
all but a finite number of the
$d_j$ being zero.
(2) The representation is unique
for all positive integers
if and only if
$B_j$ divides $B_{j+1}$
for all $j$.
Uniqueness then holds
in the case mentioned above
($B_j = b^j$),
but also holds in the factorial numbering system,
where
$B_j = (j+1)!$.
