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If you want to declare a setter, you must set the property equal to the incoming value.

public String userInput
{
    get { return userInput; }
    set { userInput = value; }
}

You can also use the simplified form here since you are not doing anything more fancy:

public String userInput { get; set; }

@sfdcfox has astutely pointed you towards Building a Custom Controller from the Visualforce Developer Guide, but I think his own explanation is actually a better read for you here: Apex Controllers: What do the get; set; do?