Is $\partial (\liminf_{n}\sum_{t=1}^{n}a_{t})/\partial a_{t} = 1$ for every $t \in \mathbb{N}$?

Question

If we do not impose any conditions on the real sequence $(a_{t})$ other than that $a_{t} \geq 0$ for all $t$, is it necessarily true that for every $t \in \mathbb{N}$ we have $\partial (\liminf_{n}\sum_{t=1}^{n}a_{t})/\partial a_{t} = 1$? The function under consideration is $(a_{1},a_{2},\dots) \mapsto \liminf_{n}\sum_{t=1}^{n}a_{t}$.

Answer 1

Because $\liminf_{n}\sum_{t=1}^{n}a_{t}$ doesn't depend on any particular $a_n$, it only depends on the sequence as a whole, we get that

$$\partial (\liminf_{n}\sum_{t=1}^{n}a_{t})/\partial a_{n} = 0$$