Hi, I'm currently a undergrad student and the story is that there is this professor from another university which I did some summer research with him, and I guess we had a pretty nice relationship. I sent emails just to greet him and share interesting things from time to time, and he'd reply to them just like a friend. Last time, he even wrote a reference letter for my application for exchange programme. However, about 4 days ago, I sent another email asking him about some stuff for grad school application, e.g. tips for research proposals, reference letter...etc., and I haven't heard anything from him yet. I know 4 days might not be very long, but I'm pretty sure he checks his email frequently and my past experience was that he usually replies within 2 days. I'm very anxious right now, could anyone share some opinion on this? Also, if he doesn't reply in a week(seems to be very possible right now), I'm planning to write a follow-up letter to remind him. Will that be impolite or even rude? If not, how should I phrase it up?