I have come across that if a sequence of real numbers, say $x\_n$ which is monotonically increasing and converges to a real number $x$ then $x\_n \leq x$ for all $n\geq 1$. Is this statement is correct? If correct then why , according to me suppose if $x\_1>x$ then since X-n is monotonically increasing so, $x\_2\geq x\_1$ so, distance between $x\_2$ and $x$ is more than $x\_1$ and $x$, similarly all terms will go further and further from x , and if we choose $\epsilon$ to be $d(x\_1,x)/2$ then no terms of the sequence will be in this $\epsilon$- neighborhood of x which is a contradiction/ AM I right ?? Is there any rigorous proof of this??