For any two n-dim vector $v$ and $v'$ define $v\leq v'$ iff for each $1\leq i\leq n$, $v\_i\leq v\_i'$. 
Suppose further that the entry of vectors can only take values from $m$ distinct values $\{a\_1, a\_2, \cdots, a\_m\}$. 


Claim: for a sequence of vectors $v\_1\leq v\_2 \leq \cdots \leq v\_s$ for $s=mn+1$, we must have some $i,j$ such that $v\_i=v\_j$. 


This seems to be trivial, but is there any formal and simple proof for that? Or does this follow from some famous theorem (e.g. some Erdos theorem)?



Monotonic sequence (edited)
I guess the OP means that the values are ordered linearly, i.e. $a\_1<\dots<a\_m$. In that case the statement is even true for $s=mn-n+2$. 


To see this, define the numbers $1\leq c(i,k)\leq m$ such that $v\_i=(a\_{c(i,1)},\dots,a\_{c(i,n)})$. In addition, consider the integers $w\_i:=c(i,1)+\dots+c(i,n)$. Then, for any pair $1\leq i<j\leq s$, we have $v\_i\leq v\_j$, hence $c(i,k)\leq c(j,k)$ for any $1\leq k\leq n$, hence also $w\_i\leq w\_j$. Moreover, $w\_i=w\_j$ can only hold when $c(i,k)=c(j,k)$ for each $1\leq k\leq n$, i.e. when $v\_i=v\_j$. 
Now observe that $n\leq w\_1\leq\dots\leq w\_s\leq mn$, hence for $s=mn-n+2$ we have $w\_i=w\_j$ for some $i<j$, so $v\_i=v\_j$ by the discussion above. The proof is complete.


