Why is $|1-a|+|a-(-1)| < 1+1$?
This is for proving that $(-1)^n$ does not converge (for this prove you have to use the triangle inequality).
You have to assume that $\lim\limits_{n->∞} (-1)^n = a$, and $a$ is a real number.
quote:
2 = |1 −(−1)| = |1 −a+a−(−1)| ≤ |1 −a|+|a−(−1)| < 1+1= 2. This absurdity shows our assumption lim(−1)^n = a must be wrong, so the sequence (−1)^n does not converge.
This is false.
The correct statement is
$$|1-a| + |a-(-1)| \ge 2$$
To see it, notice that $|1-a|$ is the distance of $a$ from $1$ and $|a-(-1)|$ is the distance of $a$ from $-1$. If $-1 \le a \le 1$, it is equal to $2$.
For $a < -1$, its distance to $1$ is clearly more than $2$.
For $a > 1$, its distance to $-1$ is clearly more than $2$.
This also implies that $(-1)^n$ diverges as it's distances from $1$ and $-1$ can't be arbitrarily small simultaneously. More generally, for a sequence to converge, you can't have two subsequences that converges to two different values.