Quantifier

Let S[1], .... , S[n] be a sequence satisfying
(a) S[1] is a positive integer and S[n] is a negative integer
(b) For all i, 1 <= i < n, S[i+1] = S[i] + 1 or S[i+1] = S[i] -1
Prove that these exists i, 1 < i < n such that S[i] = 0
ANSWER
Assume that these does not exist i, 1 < i < n such that S[i] = 0
First,we prove that for every 2<=i<=n-1,S[i] >=1
when n=2
S[2]=S[1] + 1=2 or S[2] = S[1] -1=0 (rejected)
prove that S[2] >=1
now assume that when 2<=k<=n-2,S[k] >=1
S[k+1]=S[k] + 1 or S[k]-1
Since S[k] >=1, and S[k+1] does not equal to 0
We can deduce that the minimum value of S[k+1] is 1
(occurs when S[k]=2 and S[k+1]= S[k]-1)
So for all 2<=i<=n-1,S[i] >=1
But S[n] = S[n-1] + 1 or S[n] = S[n-1] -1
Since S[n-1]>=1, the the minimum value of S[n] is 0
Contradict the fact that S[n] is a negative integer
So we conclude that these exists i, 1 < i < n such that S[i] = 0