Suppose we can enumerate them, for example,
$\displaystyle \left< \boxed{4}, 5, 11, 19, 121, ... \right>$
$\displaystyle \left< 2, \boxed{7}, 12, 13, 99, ... \right>$
$\displaystyle \left< 7, 8, \boxed{9}, 10, 12, ... \right>$
$\displaystyle \left< 1, 2, 99, \boxed{100},222, ... \right>$
...
Define a new sequence:
$\displaystyle \left< 4+1,7+4,9+7,100+9, ... \right>$
I hope you can see how we defined this sequence? The first term is the boxed term +1, then the second term is the last term plus the second boxed term, and so on. This creates a new increasing sequence which is now found on this list.
(This was the diagnol argument)