Suppose that $a_n=O(n)$ and $b_n=O(n^2)$. Let $c_n=a_n-b_n$

Question

Suppose that $a_n=O(n)$ and $b_n=O(n^2)$. Let $c_n=a_n-b_n$.

Prove that $c_n=O(n^2)$ and not necessarily $c_n=O(n)$

$$u_n=O(v_n)\Leftrightarrow \exists L\geq0, \exists \overline n, \forall n>\overline n;|u_n|\geq L|v_n| $$

I came up with:

1)Let $L_1,L_2\geq0$

$$|c_n|=|a_n-b_n|\leq|a_n|+|b_n|=L_1\cdot n+L_2\cdot n^2$$

Let $L_3=max\{L_1,L_2\}$. then:

$$|c_n|\leq L_3(n+n^2)$$

\end

And I have no idea what to do with this(because I should get $c_n\leq L_3(n^2)$)

Answer 1

Note that $n+n^2=n(n+1)\leq n\cdot 2n=2n^2$. Now let $L_4=2L_3$, and you're done.

As for $c_n$ not necessarily being $O(n)$, think about what happens if $a_n=0$ and $b_n=n^2$.