If we assume |g(y)| > 1 and c is the lup (least upper bound) of |f(x)|, then if f(x) !=0, we can say 2c >=|f(x+y)| + |f(x-y)|. The right side of this inequality is simply 2 * |f(x)|*|g(y)|. So, 2c >= 2 * |f(x)| * |g(y)|. Rearranging this, one achieves, c/|g(y)|>=|f(x)|. But, we assumed |g(y)| >1. So, C/|g(y)| is less than 1, since the lup of |f(x)| can be maximum 1 if f(x) = 1 or f(x) = -1, according to the given |f(x)|<=1. Remember that it was assumed to be |g(y)| >1, so c/|g(y)| < c. If you divide something positive by a positive number greater than 1, than the thing being divided is bigger than the result. So, this would make c/|g(y)| the least upper bound, but this is smaller than c. So, this is a contradiction. So, we can say |g(y)| <=1 for the case f(x)!=0.