I'm using the after update trigger to update the "BillingCity" field in the Account object and make the field in proper case.
 

trigger on Account (after update) {

            String billingCity = Account.billingCity;

            toTitleCase(billingCity);
}

And the code in the invoked toTitleCase(billingCity) method:
 

//converts a given string to Title Case where the
//first letter of every word is capitalised and the rest are small
public String toTitleCase(String billingCity) {
   System.debug('billingCity ' + billingCity);
   String titlePhrase = '';
   //a set of words that should always be in lower case when in Title Case
   Set<String> forceLower = new Set<String>{'of', 'the', 'for', 'and', 'a', 'to', 'at' ,'an', 'but', 'if', 'or', 'nor'};

   if(billingCity != null && billingCity.length() > 0){
      String[] splitPhrase = billingCity.trim().split(' ');

      for(integer i = 0; i < splitPhrase.size(); i++){
          if(!forceLower.contains(splitPhrase[i].toLowerCase()) || i == 0 || i == (splitPhrase.size()-1) ) {
             titlePhrase += (splitPhrase[i].substring(0,1).toUpperCase())+(splitPhrase[i].substring(1).toLowerCase())+' ';

          } else {
               titlePhrase += splitPhrase[i].toLowerCase()+' ';

               }
          }
       }

       return titlePhrase.trim();
}

In the debug log I can see that the "BillinCity" field is updated with the proper case:
 

17:00:57:945 USER_DEBUG [40]|DEBUG|billingCity San Marino

but on a record level the field remains with upper case: "SAN MARINO"

Please advise an appropriate solution.