A sequence close to a decreasing sequence is nearly decreasing ?

Question

Suppose we have two real sequences such that $\vert x_n - y_n\vert \le {(\frac12)} ^n$ and that $x_n \downarrow 0$.

This doesn't mean that $y_n$ is decreasing. But is it true that $y_n +2^{2-n}$ will also decrease to 0 ?

Answer 1

$(y_n + 2^{2-n})_n$ converges to zero as $n$ tends to infinity, but it's not necessarily decreasing.

• We have $x_n \to 0$ and $(y_n-x_n) \to 0$, so $y_n = (y_n - x_n) + x_n \to 0$ as $n$ tends to infinity.
• Since $0 < 2^{-1} < 1$, $2^{2-n} = 4 \cdot 2^{-n} \to 0$ as $n$ tends to infinity.
• Thus, $(y_n + 2^{2-n})_n \to 0$ as $n$ tends to infinity.

To see why $(y_n + 2^{2-n})_n$ is not necessarily decreasing, take a counterexample: $x_n := 1/n$, $a_n := (-1)^{n+1} 2^{-n}$, $y_n := x_n + a_n$ so that $|x_n - y_n| = |a_n| = 2^{-n}$.

1. $y_1 = 1+1/2 = 3/2$
2. $y_2 = 1/2-1/4 = 1/4 < 3/2$
3. $y_3 = 1/3+1/8 = 11/24 > 1/4$