Here is the full question:
Consider a random walk where we start at the number 0 and in each step we move from x to x+1 or x-1 with equal probability. Prove that we eventually return to 0 with probability 1.
And here is my proof:
The only way that we do not return to the origin is if we continue stepping to the right or left infinitely.
However, because stepping is an independent event, we multiply the probabilities, and realizing this we can say that $$ \lim_{x\to\infty} (\frac{1}{2})^x = 0. $$
In other words, the probability of not returning to the origin is 0. So the probability of returning to the origin is 1.
QED
Does this make sense? After researching random walks and it's complexity I feel like this is not strong at all, but I am not sure why. Or is it?
You have given the probability of two of the ways of not returning to the origin: step right and then keep stepping right forever, or step left and then keep stepping left forever.
But there's also step right twice, then left, then right forever. Or step right three times, then left twice, then right forever. Or right twice, left once, repeat forever. Or right $n$ times, left $m < n$ times, then right $n'$ times, left $m' < n' - m + n$ times, and so forth.
In fact if you just consider the possible sequences of step right twice then left once, or step right three times than left once, continue forever with any combination of two-step and three-step sequences, there are uncountably many complete sequences, so even though each complete sequence has probability zero you can't be sure that their total probability is zero. (The set of all infinite sequences of steps also is composed of uncountably many sequences of probability zero each, but the total probability of that set is $1.$)
That's why you need a more sophisticated approach.