Why does price equal marginal revenue for a firm in a perfectly competitive industry?

Why does price equal marginal revenue for a firm in a perfectly competitive industry?
In a perfectly competitive industry marginal revenue or (the cost to produce one more unit) stays constant
so for example a pencil costs 1 dollar to make at the 101st pencil it will still cost 1 dollar to make. the price at which it must sell it at is also one dollar because if the company decides to raise the price it will lose all of its consumers to another firm competing with them that sells pencils at 1 dollar. the firm would be able to sell nothing at a higher price because the market is so competitive
therefore, you can not raise the marginal revenue without raising the price and you cannot raise the price because the firm runs the risk of selling nothing therefore they stay equal.

A perfectly competitive firm takes the market price as given, (They cannot set the price at which they sell the item the other firms through supply and demand have already sorted that out) so the firm-specific demand curve is horizontal. The firm can sell all it wants at the market price, but would sell nothing if it charged a higher price.