Your finger size may change over time and the wedding band or the engagement ring may also need to be resized.
Sometimes the engagement ring and wedding band come together, but whether or not it does or doesn't you can certainly show her the wedding band.
There are most of the time two reasons. Either they do not want to loose the wedding ring or They do not want wear to much jewelry to appear flashy.
His wedding
If he has a wedding ring then yes. If he has a wedding bracelet or necklace in it's place then they would be his wedding band. If it is a ritualistic tattoo, piece of clothing or (girls only) red dot on the forehead then they would not be considered wedding bands.
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The ancients did not use wedding rings. The Romans sometimes would give their intended brides an iron ring, but this was not mandatory. On occasion, a wealthy couple would give each other rings, but as gifts, not as wedding rings.
Yes, a man should buy two rings. Sometimes the groom buys both rings, sometimes the couple pays for them together and sometimes they are considered part of the wedding expense and the bride's father pays for them. The man buys the engagement and wedding ring for his wife and the woman buys the wedding ring for her husband.
My wedding set (engagement ring and matching wedding band.)
A wedding ring might swell if it were injured. Arthritis or an infection might also cause swelling.
The groom should buy the engagement and wedding ring for his bride to be and the bride to be should buy the groom's wedding band.
A ring.. but NOT a wedding ring
For some women they may take off their wedding rings when housecleaning as different chemicals could ruin the rings. For men, the only time they would need to take off their ring is if they are working with machinery and the ring could catch on something. If the ring is too loose or too tight take it to a jewelry and the cost for having it sized is minimal.