For example, how could I prove that the following language is deletion closed:{$a^k$$b^j$ : $j$, $k$ $\geqslant$ 0}The reason seems obvious to me, I just can't see a way to prove it. 
How do I prove that a language is deletion closed?
formal languages;regular languages;proof techniques;closure properties
You'd prove that a language $L$ is deletion-closed as follows.Consider a word $w\in L$ and let $w = w_1\dots w_\ell$.  Now, for any $r$ and $s$ with $1\leq r\leq s\leq \ell$, consider the word $w' = w_1\dots w_{r-1}w_{s+1}\dots w_\ell$, i.e., the word that results from deleting the substring $w_r\dots w_s$ from $w$. We have $w'\in L$ because [some argument that will be specific to the language $L$ you're working with].