Unobtrusive validation doesn’t work with Ajax.BeginForm

That’s because the second view is loaded with AJAX at a later stage and you need to call $.validator.unobtrusive.parse(...)
immediately after its contents is injected into the DOM in order to enable unobtrusive validation. Look at the following blog post for more details.

So in your case, instead of alerting in the OnSuccess callback of the first AJAX call, subscribe to a javascript function which will invoke this method:

@using (Ajax.BeginForm(
    "Action1",
    "Controller",
    null,
    new AjaxOptions { 
        OnSuccess = "onSuccess",
        UpdateTargetId = "result"
    },
    null)
)
{
    <input type="submit" value="Save" />
}

and then in your javascript file:

var onSuccess = function(result) {
    // enable unobtrusive validation for the contents
    // that was injected into the <div id="result"></div> node
    $.validator.unobtrusive.parse($(result));
};