# Math Help - Real Analysis - Squeeze Theorem

1. ## Real Analysis - Squeeze Theorem

Show that if xn <= yn <= zn for all n Є N, and if lim xn = lim zn = b, then lim yn = b.

2. Originally Posted by ajj86
Show that if xn <= yn <= zn for all n Є N, and if lim xn = lim zn = b, then lim yn = b.
Let $\epsilon > 0$ be given. since $\lim x_n = b$, there exists an $N_1 \in \mathbb{N}$ such that $n > N_1$ implies $|x_n - b| < \epsilon$. Similarly, there is $N_2 \in \mathbb{N}$ such that $|z_n - b| < \epsilon$ whenever $n > N_2$.

now, take $N = \text{max} \{ N_1,N_2 \}$, and assume $n > N$. Note that since $x_n \le y_n \le z_n$, we have $x_n - b \le y_n - b \le z_n - b$....

can you finish?