Need clarification due to confusing wordings of over question concerning infinite sequences I am trying to figure out what I am suppose to prove for the following question on infinite sequences.  The wording is confusing for me.

A real number $x$ is irrational, if we can find an ascending sequence of integers $(q_n)$, such that $xq_n$ is not an integer for any n, but if, when $p_n$ stands for the integer nearest to $xq_n$, $(xq_n - p_n)$ is a null sequence.

Is this an if and only if type question or am I asked to prove that $x$ is irrational or that $(xq_n - p_n)$ is a null sequence.
Thank you in advance
 A: There’s only one “if” in the statement of the proposition. You only need to show the implication in one direction.
Specifically, consider the part of the statement that comes after the word “if”. 
If all of that is true, then $x$ is irrational. That’s what you have to prove. 
A: I don't know whether English is your native language, but I can certainly see why this could cause confusion for a non-native speaker, and probably for some native speakers as well, if they are unfamiliar with mathematical verbiage. Below is my attempt to make this as explicit and unambiguous as I can. Note that I've included a special case in the definition of $p_n,$ which won't matter in the long run (indeed, for each $n$ you can choose either the lesser or greater closest integer), but certainly in the short run, not including something like this creates some ambiguity.
Let $x$ be a real number and let $(q_n)$ be a strictly increasing sequence of integers. Also, for each positive integer $n,$ let $p_n$ be the integer closest to $xq_n$ (if there are two such closest integers, let $p_n$ be the greater of the two). Assume that both (1) and (2) hold: (1) for each positive integer $n,$ the real number $xq_n$ is not an integer; (2) $(xq_n - p_n) \rightarrow 0$ as $n \rightarrow \infty.$ Prove that $x$ is irrational.
