Let ${x_n}$ be an increasing sequence of irrational numbers in $[0,2]$. Then

Question

(a) ${(x_n)}$ converges to $2$

(b) ${(x_n)}$ converges to ${\sqrt 2}$

(c) $(x_n)$ converges to some number in $[0,2]$

(d) $(x_n)$ may not converge.

Now I took the example of the sequence ${\frac{\sqrt n}{\sqrt {n+1}}}$ and it converges to $1$. So I ruled out options (a) and (b). Is there any example that confirms option (d) ?

Answer 1

The $x_n$ are increasing and bounded above. Hence, the sequence converges to its supremum.