As an addition to the infinitive: In general, infinitive- sentences with "zu" always must be formed whenever the verb from the main clause refers to the action of the second clause. for example: Ich versuche, das Tor zu treffen: --> the action "versuchen" refers to the action "treffen", so you have to put "zu" in front of it. and it is the same in your first sentence: Sie hört auf zu weinen. --> the action "aufhören" refers to "weinen", so "zu" must be in front of "weinen". All verbs, which necessarily refer to an action, use the infinitive: aufhören, beginnen, anfangen, verbieten, vergessen, ... And also all verbs, which can be used with a "dass"-sentence: glauben, denken, meinen, hoffen, wissen,... I hope it helps =)