Is this a valid proof of Hall’s Theorem on System of Distinct Representatives?

Hall’s Theorem states that for sets $ A_1,…,A_n$ if and only for all $ J\subseteq \{1,…,n\}$ we find $ |\bigcup_{j\in J}A_j|\geq|J|,$ then we can choose $ x_i\in A_i$ for all $ i$ such that $ \{x_1,…,x_n\}$ is a system of distinct representatives for $ A_1,…,A_n.$ To me it seems easiest to some inductive argument (or a contradiction that is similar to induction), however, in my graph theory class we’re working with Flow Algorithm’s, so the goal is to solve it using the Max-Flow-Min-Cut Theorem. I’ve seen one proof using that theorem, but I think I’ve came up with another, which to me seems more natural. I was wondering if my proof is correct. Here’s the proof.

$ \textbf{Proof:}$

Throughout $ x$ shall refer to an element of some $ A_i.$ We set up a capicated network as follows $ $ \overrightarrow{E}=\{(s,A_i):\forall A_i\}\cup\{(A_i,x):\forall A_i\text{ and }x\in A_i\}\cup\{(x,t):\forall x\in \cup_{i=1}^n A_i\}$ $ with capacities for $ (s,A_i)$ and $ (x,t)$ all $ 1$ and all other capacities infinite. This is ‘depicted’ in the following link, where any unwritten capacity is infinite.

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Now we suppose that we have a minimal network cut $ U.$ So the sum of capacities of edges leaving $ U$ is minimal. If $ A_i\in U,$ and $ x\not\in U$ for sum $ x\in A_i,$ then $ (A_i,x)$ is an edge leaving $ U$ with infinite capacity, and $ \{s\}$ has smaller capacity than $ U,$ impossible. So $ x\in U$ for all $ x\in A_i$ . We compute $ $ \text{capacity}(U)=\sum_{A_i\not\in U}\text{capacity}(s,A_i)+\sum_{x\in U}\text{capacity}(x,t)=|\{A_i\}\backslash U|+|\{x:x\in U\}|$ $ $ $ \text{capacity}(U)\geq|\{A_i\}\backslash U|+|\{x\in A_i: A_i\in U\}|\geq|\{A_i\}\backslash U|+|\{A_i\in U\}|=n.$ $ Thus since $ \text{capacity}\{s\}=n$ we find that $ \{s\}$ is a min cut, and we can find a system of distinct representatives. $ \blacksquare$

ping at University halls of residence

One computer has a ssh server running on Linux, and I can login on the localhost with : ssh myuser@localhost.

The Linux computer IP is : 10.188.116.102

The other computer has Windows 7.

The Windows computer IP is : 10.188.40.210.

Now, I’m trying to connect to the Linux computer from the Windows computer, both on the desk, but ping has negative answer :

either

ping -t 10.188.40.210 

from the Linux computer, or

ping -t 10.188.116.102 

from the Windows computer answer :

Destination Host Unreachable


Also, here is the result of ifconfig on the Linux computer :

enp0s25: flags=4163<UP,BROADCAST,RUNNING,MULTICAST>  mtu 1500         inet 10.188.116.102  netmask 255.255.0.0  broadcast 10.188.255.255         inet6 fe80::6ef0:49ff:fe2a:8dc9  prefixlen 64  scopeid 0x20<link>         ether 6c:f0:49:2a:8d:c9  txqueuelen 1000  (Ethernet)         RX packets 171603  bytes 242463049 (231.2 MiB)         RX errors 0  dropped 0  overruns 0  frame 0         TX packets 63232  bytes 6620049 (6.3 MiB)         TX errors 0  dropped 0 overruns 0  carrier 0  collisions 0         device interrupt 16  memory 0xfc200000-fc220000 

and the some network details on the Windows computer :

IP   : 10.188.40.210 mask : 255.255.0.0 DHCP : 10.188.0.1 

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