Originally Posted by

**koko2009** Let

{*xn*} be a sequence of real numbers that is bounded above by *M *and such that

*xn *® *x *. Prove that *x *£ *M *. we know that A

*sequence *{*xn*} is *bounded *if there exists an *x *Î*X *and a real number

*M *> 0 such that *d *(*xn *, *x*) £ *M *for all *n *. also it convergent to x.

how can I use this to prove. any help will be apprciated.

$\displaystyle x>M \Longrightarrow\,\, choose\,\, e=\frac{x-M}{2}\Longrightarrow |x_n-x|<e\,\,\, \forall n>N\,,\,\,for\,\,some\,\,N \in \mathbb{N}$ $\displaystyle \Longrightarrow \frac{M-x}{2}= -e \leq x_n-x \leq e=\frac{x-M}{2}$

Now get your contradiction that $\displaystyle \forall n \in \mathbb{N}\,,\,|x_n|<M$

Tonio