Show that each $k\in \mathbb{N}$ can be represent as $ k = S_m(a)+S_n(b)$ Let $a,b$ are non negative integers and $m,n$ are positive integers.
Let define 
$$S_m(a)=0^m+1^m+2^m+...+a^m$$
and $$S_n(b)=0^n+1^n+2^n+...+b^n$$

Question

Show that each $k\in \mathbb{N}$ can be represent as
$$ k = S_m(a)+S_n(b)$$


Example
$5= S_2(2)+S_n(0)$
$6= S_1(3)+S_n(0)$
$\ \ = S_1(2)+S_1(2)$
Edit
I just changed whole question to simplify , understanding and more motivation, my apologies.
I already asked for representation of $ k = S_1(a)-S_1(b)$ here
 A: It appears that this claim is untrue. There are quite a few numbers which cannot be expressed in this manner, the smallest being $52$.
Observe the terms $S_n(k) \leqslant 52$
First powers = $\{0,1,3,6,10,15,21,28,36,45\}$
Second powers = $\{0,1,5,14,30\}$
Third powers = $\{0,1,9,36\}$
Fourth powers = $\{0,1,17\}$
Fifth powers = $\{0,1,33\}$
Rest of the powers = $\{0,1\}$
Union of all such values = $\{0,1,3,5,6,9,10,14,15,17,21,28,30,33,36,45\}$
You can easily check that for all values $m$ in the union set, $52-m$ is not in the union set. Thus, $52$ cannot be represented in the required manner. The first few values of this sort are:
$$52,77,89,116,118,147,152,179,187,197,...$$
This can be easily coded using the logic executed above for $52$.
S=[0]
i=1
while i<=10:
    add=0
    j=1
    while j<=50:
        add=add+(j**i)
        S.append(add)
        j=j+1
    i=i+1
print(S)
for m in range(1,1000):
    sm=0
    for index in range(len(S)):
        if m-S[index] in S:
            sm=sm+1
    if sm==0:
        print(m)

