Let (xn) be a convergent sequence, and let a, b ∈ R.

If a≤x_n ≤b for all n,then a≤ lim x_n ≤b.

First we show a≤x_n. Let y_n = x_n - a

If x_n ≥ a for every n then y_n ≥ 0 for every n

So as proven already in another problem prior to this (which for my formal proof i will write up) lim y_n ≥ 0 which implies lim x_n ≥ a. Next Show x_n ≤b

Let y_n = -(x_n - b) so y_n ≥ 0 for every n and we can reach the same conclusion

lim y_n ≥ 0 which is lim -x_n + b ≥ 0 and finally lim x_n ≤ b