I was watching the following video on Random Walks: https://www.youtube.com/watch?v=iH2kATv49rc
Here, the following point is discussed:
- Suppose you have an infinite square grid.
- In each "step", you can move one "step" in the Up, Down, Left or Right Direction
- Suppose you are at the origin at step = 0
- Given this information, is only possible to return to the origin in an even number of steps
My Question: While I think this is obvious, I was interested in knowing how statements like this can be mathematically proven.
I think this can be proven using mathematical induction ($n$ is the number of steps).
- When $n$ = 0, you are at the origin
- Assume that in $2n$ steps you can return to the origin: And $2n$ is an even number
- Now consider $2n+2$ steps : $2n+2$ is also an even number
- Since you can only return in an even number of steps and $2n+2$ is an even number - the proof is concluded.
Is this sufficient to conclude the proof?
Thanks!
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