2: Proof by definition:
an = 3n + (−1)n ≤ 3n + 1 < 3(n + 1) − 1 ≤ 3(n + 1) + (−1)n = an +1.
The function approach cannot be used since one cannot define
f (x) = 3x + (−1) x.
Indeed, general powers cannot be defined for negative bases.
Next hint - boundedness Answer