Deterministicaly stops winning, yet if conditioned on time there is positive probability to win in the future.

Question

In the answer to question https://mathoverflow.net/questions/12462/limsup-and-liminf-for-a-sequence-of-sets "has2" gives a concrete example for the use of $\limsup$ and $\liminf$ in form of a "card game". I quote the example below. What I ask is unrelated to the original question, and here it is:

Intuitively, how can it be that there, deterministically, is a time after which the player in the game stops winning. Yet, conditioned at any point in time (i.e. step $n$ of the game) there is always positive probability to win in the future?

I can not, intuitively, settle those two facts together, is there any re-phrasal of those two statements which makes (more) evident they do not contradict one another?

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Here is a conceptual game that can be partially understood using these concepts: We have a deck of cards, on the face of each card an integer is printed; thus the cards are $\{1,2,3...\}$. At the nth round of this game, the first $n^2$ cards are taken, they are shuffled. You pick one of them. If your pick is $1$, you win that round. Let $A_n$ denote the event that you win the nth round. The complement $A^c_n$ of $A_n$ will represent that you lose the $n$-th round. The event $\limsup A_n$ represents those scenarios in which you win infinitely many rounds. The complement of this event is $\liminf A^c_n$, and this represents those scenarios in which you ultimately lose all of the rounds. By the Borel Cantelli Lemma $P(\limsup A_n) =0$ or equivalently $P(\liminf A^c_n)=1$. Thus, a player of this game will deterministically experience that there comes a time, after which he never wins.

(emphasis added)

Answer 1

The positive probability of winning decreases sufficiently fast as time passes.

For example, at time 1, probability of winning is 1/2. Time 2, 1/3. Time 3, 1/4. And so on.

Of course

$$\lim H_n = \infty$$

so that is not sufficiently fast.

However, if we have 1, 1/4, 1/9, 1/16, ... that is decreasing sufficiently fast.

Of course in precise terms when we say decreasing sufficiently fast we mean that

$$\sum_{n=1}^{\infty} \frac{1}{n^2} < \infty$$

In general:

$$\sum_{n=1}^{\infty} p_n < \infty$$

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Your question is analogous to the Dichotomy Paradox of Zeno:

Suppose Homer wants to catch a stationary bus. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a quarter, he must travel one-eighth; before an eighth, one-sixteenth; and so on.

...

This description requires one to complete an infinite number of tasks, which Zeno maintains is an impossibility.