Question 3.T.6: Suppose the sequences (xn), (yn), and (zn) satisfy xn ≤ yn ≤...

Given ε > 0, there are $N_{1}, N_{2} ∈ \mathbb{N}$ such that

$|x_{n} − \ell| < ε$  for all $n ≥ N_{1}$

$|z_{n} − \ell| < ε$  for all $n ≥ N2$.

Now we define N = max$\left\{N_{0}, N_{1}, N_{2}\right\},$ and note that

⇒ $x_{n} ≤ y_{n} ≤ z_{n},  \ell − ε < x_{n} < \ell + ε,  \ell − ε < z_{n} < \ell + ε$

⇒ $\ell − ε < x_{n} ≤ y_{n} ≤ z_{n} < \ell + ε$

⇒ $\ell − ε < y_{n} < \ell + ε$

⇒ $| y_{n} − \ell| < ε$,

which means $y_{n} → \ell$.