If someone did wudu once in a day and prayed all the five prayers with just that one wudu, is it permissible even if he did not do anything to break the wudu?
It's totally permissible if he did not do anything to break the wudu. However, one reason of getting wudu before Solat is to get ready mentally and spiritually for Solat. Because of this reason it's better to repeat it every time you want to pray. Addition to that you might've been broken your wudu without realizing.
“He who performs the Wudu perfectly (i.e., according to Sunnah), his sins will depart from his body, even from under his nails.”
Make sure you follow your heart and not your desires and always strive for best. Allah knows your best under the circumstances you are in.
Benefits of wudu: Link
Allah knows the best !