Though the mind in its reasonings from causes or effects carries its view beyond those objects, which it sees or remembers, it must never lose sight of them entirely, nor reason merely upon its own ideas, without some mixture of impressions, or at least of ideas of the memory, which are equivalent to impressions. When we infer effects from causes, we must establish the existence of these causes; which we have only two ways of doing, either by an immediate perception of our memory or senses, or by an inference from other causes; which causes again we must ascertain in the same manner, either by a present impression, or by an inference from their causes, and so on, till we arrive at some object, which we see or remember. It is impossible for us to carry on our inferences IN INFINITUM; and the only thing, that can stop them, is an impression of the memory or senses, beyond which there is no room for doubt or enquiry.
Enquiry Concerning the Principles of Morals, Appendix 1:
It appears evident that—the ultimate ends of human actions can never, in any case, be accounted for by reason, but recommend themselves entirely to the sentiments and affections of mankind, without any dependance on the intellectual faculties. Ask a man WHY HE USES EXERCISE; he will answer, BECAUSE HE DESIRES TO KEEP HIS HEALTH. If you then enquire, WHY HE DESIRES HEALTH, he will readily reply, BECAUSE SICKNESS IS PAINFUL. If you push your enquiries farther, and desire a reason WHY HE HATES PAIN, it is impossible he can ever give any. This is an ultimate end, and is never referred to any other object.
Perhaps to your second question, WHY HE DESIRES HEALTH, he may also reply, that IT IS NECESSARY FOR THE EXERCISE OF HIS CALLING. If you ask, WHY HE IS ANXIOUS ON THAT HEAD, he will answer, BECAUSE HE DESIRES TO GET MONEY. If you demand WHY? IT IS THE INSTRUMENT OF PLEASURE, says he. And beyond this it is an absurdity to ask for a reason. It is impossible there can be a progress IN INFINITUM; and that one thing can always be a reason why another is desired. Something must be desirable on its own account, and because of its immediate accord or agreement with human sentiment and affection.
Hume has other arguments that appeal to infinite progressions -- infinite diminution arguments based on the idea that a finite is exhausted to nothing by infinite diminution, or arguments that justification will be inadequate even if it is carried on infinitely; but these are, as far as I am aware, his only explicitly espoused infinite regress arguments, taken in the proper sense.
