Does this limit necessarily exist?

Question

Let $a_{t} = a_{t-1}+1$ for all $t \geq 1$; let $a_{0}$ be given. Then we can write $a_{0} = a_{1} - 1 = a_{2}-1-1 = a_{2}-2 = \cdots = x$. It is tempted to write $x = \lim_{t\to\infty}(a_{t}-t)$ (and such a thing is done not infrequently in an applied math; macroeconomics, for example.). But I wonder if this is legit? Should we write $x = \liminf_{t\to\infty}(a_{t}-t)$ instead for safety?