Prove that the ant can survive

Question

There is a table with infinite cells. An ant starts from cell $(1,1)$ and each time it can move one cell up or right. Before starting to move, an infinite sequence of cell numbers like $<(x_{1},y_{1}) , (x_{2},y_{2}), ... , (x_{n}, y_{n}) , .... >$ is given to it. After step $k$, cell number $(x_{k},y_{k})$ will be poisoned and if ant goes there or already is there, it'll die. Prove with induction that the ant can live forever if it knows sequence elements from the beginning :)

Answer 1

Hint: Consider for each level of the graph the number of possible cells at that level the ant can reach alive, minus the number of cells on later levels that are already poisoned at that time.

Show that this number cannot decrease.

Conclude that there are reachable cells on every level.