Suppose the sequences (xn), (yn), and (zn) satisfy xn ≤ yn ≤ zn for all n ≥ N0. If lim xn = lim zn = l, then (yn) is convergent and its limit is also l.

Question

Suppose the sequences [latex](x_{n}), (y_{n}),[/latex] and [latex](z_{n})[/latex] satisfy

[latex]x_{n} ≤ y_{n} ≤ z_{n}[/latex] for all [latex]n ≥ N_{0}[/latex].

If lim [latex]x_{n} =[/latex] lim [latex]z_{n} = \ell,[/latex] then [latex]( y_{n})[/latex] is convergent and its limit is also [latex]\ell[/latex].

Accepted Answer

Given ε > 0, there are [latex]N_{1}, N_{2} ∈ \mathbb{N}[/latex] such that

[latex]|x_{n} − \ell| < ε[/latex] for all [latex]n ≥ N_{1}[/latex]

[latex]|z_{n} − \ell| < ε[/latex] for all [latex]n ≥ N2[/latex].

Now we define N = max[latex]\left\{N_{0}, N_{1}, N_{2}\right\},[/latex] and note that

[latex]n ≥ N ⇒ n ≥ N_{0}, n ≥ N_{1}, n ≥ N_{2}[/latex]

⇒ [latex]x_{n} ≤ y_{n} ≤ z_{n}, \ell − ε < x_{n} < \ell + ε, \ell − ε < z_{n} < \ell + ε[/latex]

⇒ [latex]\ell − ε < x_{n} ≤ y_{n} ≤ z_{n} < \ell + ε[/latex]

⇒ [latex] \ell − ε < y_{n} < \ell + ε[/latex]

⇒ [latex]| y_{n} − \ell| < ε[/latex],

which means [latex]y_{n} → \ell[/latex].

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