How can I use Cross Apply to multiple rows?

I have this table :

;WITH cte AS (
    SELECT Name='john' , Times=1    
    UNION ALL
    SELECT 'paul' ,2
    UNION ALL
    SELECT 'george' , 3
    UNION ALL
    SELECT 'ringo' , 1
)

I want to display each row , Times times :

John 1
Paul 2
Paul 2
george 3
george 3
george 3
ringo 1

So I know that if I write Cross apply like :

SELECT *
FROM   cte
       CROSS APPLY(
        SELECT 1 AS ca
        UNION 
        SELECT 2
       ) y

Then each row will be displayed 2 times.

but I don't want 2 times. I want Times times

Question

How can I enhance my query to do it ?

nb :

a non-intelligent solution that came to my mind is to create a udf which create Times rows for n parameter - and then in the Cross Apply I simply do : select * from udf_toTable(Times))

;WITH cte AS (
    SELECT Name='john' , Times=1    
    UNION ALL
    SELECT 'paul' , Times=2
    UNION ALL
    SELECT 'george' , Times=3
    UNION ALL
    SELECT 'ringo' , Times=1
),
multi as
(
    select 
        Name, Times, Times as num
    from cte
    union all
    select 
        Name, Times, num - 1
    from multi 
    where num > 1
)
select Name, Times from multi
order by Name

UPDATE

without recursion

;WITH cte AS (
    SELECT Name='john' , Times=1    
    UNION ALL
    SELECT 'paul' , Times=2
    UNION ALL
    SELECT 'george' , Times=3
    UNION ALL
    SELECT 'ringo' , Times=1
)
select cte.*
from cte join 
    -- generate sequence of numbers 1,2 ... MAX(Times)
    (select top (select MAX(Times) from cte) ROW_NUMBER() over (order by object_id) rowNum from sys.objects) t
on cte.Times >= t.rowNum 
order by name