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 $\{1, 2, \cdots, m\}$.

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)?

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)?