# a sequence of real numbers

• Oct 22nd 2009, 08:05 PM
koko2009
a sequence of real numbers
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 ÎX 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.
• Oct 22nd 2009, 08:56 PM
tonio
Quote:

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 ÎX 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.

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

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

Tonio