A well-quasi-ordered set is a quasi-ordered set $(X,\leq)$ such that for any sequence $(x_n)_{n\geq 0}$ in $X$, we find $i<j$ for which $x_i\leq x_j$.
It is a well-known result that a quasi-ordered set $(X,\leq)$ is well-quasi-ordered if and only if $X$ does not contain an infinite antichain and no infinite decreasing sequence.
I don't understand the forward direction. The problem is that $\leq$ is generally not antisymmetric. Hence we can have $x_i<x_j$, while simultaneously $x_i>x_j$.
For instance, consider $X=\{0,1\}$. Say that $x\leq y$ for any $x,y\in X$. This defines a (trivial) well-quasi-ordering. Further, we find an infinite strictly decreasing sequence: $0>1>0>1>\cdots$.
Is my reasoning correct?