A monotone decreasing sequence ${x_{n}}$ converges if and only if is bounded from below
Could they please help me with this exercise?
A monotone decreasing sequence ${x_{n}}$ converges if and only if is bounded from below
Could they please help me with this exercise?
If $x_n$ is bounded from below, there exist $I = \inf x_n $. Given $\epsilon > 0$. Chose $n_0$ such that \begin{equation} I \le x_{n_0} < I+ \epsilon. \end{equation} Hence as $(x_n)$ is decreasing \begin{equation} n \ge n_0 \Rightarrow I \le x_n \le x_{n_0} < I + \epsilon. \end{equation} Then $(x_n)$ converges. Reciprocally if $(x_n)$ converges to $x$, for $n > > 1$, $x_n > x -1$ and $x_n$ is bounded below.