Finding Active Clients By Date

Question

I'm having trouble writing a recursive function that would count the number of active clients on any given day.

Say I have a table like this:

Client	Start Date	End Date
1	1-Jan-22
2	1-Jan-22	3-Jan-22
3	3-Jan-22
4	4-Jan-22	5-Jan-22
5	4-Jan-22	6-Jan-22
6	7-Jan-22	9-Jan-22

I want to return a table that would look like this:

Date	NumActive
1-Jan-22	2
2-Jan-22	2
3-Jan-22	3
4-Jan-22	4
5-Jan-22	4
6-Jan-22	3
7-Jan-22	3
8-Jan-22	3
9-Jan-22	4

Is there a way to do this? Ideally, I'd have a fixed start date and go to today's date.

Some pieces I have tried:

Creating a recursive date table

Truncated to Feb 1, 2022 for simplicity:

WITH DateDiffs AS (
    SELECT DATEDIFF(DAY, '2022-02-02', GETDATE()) AS NumDays
)

, Numbers(Numbers) AS (
    SELECT MAX(NumDays) FROM DateDiffs
    UNION ALL
    SELECT Numbers-1 FROM Numbers WHERE Numbers > 0 
    ) 

, Dates AS (
    SELECT
        Numbers
        , DATEADD(DAY, -Numbers, CAST(GETDATE() -1 AS DATE)) AS [Date]
    FROM Numbers
)

I would like to be able to loop over the dates in that table, such as by modifying the query below for each date, such as by @loopdate. Then UNION ALL it to a larger final query. I'm now stuck as to how I can run the query to count the number of active users:

SELECT
COUNT(Client)
FROM clients
WHERE [Start Date] >= @loopdate AND ([End Date] <= @loopdate OR [End Date] IS NULL)

Thank you!