One subtle assumption that needs to be made is that the receiver knows
that the sender's window size is w.

1a) Suppose the receiver receives packet i thinking that it was i-2w
and this is the smallest sequence numbered packet for which the
receiver makes a mistake.  Since packet i was sent, it must have
appeared in the sender's window.  For i to have appeared in the
sender's window, all packets up to i-w must have already been
acknowledged and the acknowledgments received by the sender.  Since i
is the smallest sequence number for which there is a mistake, the
receiver would have correctly acknowledged everything up to i-w.  If
the receiver correctly acknowledged up to i-w, then it must have known
that the sender's window had already moved to cover i-w.  At the point
in time when the sender's window covers i-w, its window has already
moved past i-2w.  Thus, the sender cannot send i-2w after it sends
i-w.  So once the receiver received i-w, it would never expect i-2w to
follow afterward.

1b) Suppose the receiver receives packet i thinking that it was i+2w
and this is the smallest sequence numbered packet for which the
receiver makes a mistake.  When the sender sent this packet i, the
largest sequence-numbered packet in the window could have been i+w-1.

If the receiver is expecting the sender to send i+2w, it must think it
has already acknowledged all packets up to i+w+1.  But the sender
cannot send i+w+1 in front of a transmission of packet i.  Hence, the
receiver must have made a mistake about packet i+w+1.  Since s=2w, the
mistake must have been for some packet i-w+1, i-3w+1, etc., but all of
these are smaller than i, contradicting the fact that i is the
smallest sequence numbered packet for which the receiver makes a
mistake.

1c) Suppose the receiver receives packet i thinking that it was i-w-1
and this is the smallest sequence numbered packet for which the
receiver makes a mistake.  Since there is no buffering, let j be the
next packet that the receiver is waiting for.  Either j<i, j=i or j>i.
In any case, since the sender sent i, the receiver must have already
ACK'd i-w and done so correctly.

j>i: then receiver has already received i, and thought it was i.  Then
the receiver must have known that the sender's window no longer
contained i-w-1, so why would it be sent later?  Contradiction

j=i: the receiver already received i-1.  Then sender's window would
have already moved past i-w-1, so why would it be sent later?  Again,
Contradiction

j<i: the receiver has already ACK'd i-w.  Were it to realize it got i,
it cannot buffer it, so it sends an ACK j-1.  If it gets i-w-1, it
does the same.  So even if the receiver makes the mistake, the
reaction is the same and the protocol proceeds correctly.

1d) Suppose the receiver receives packet i thinking that it was i+w+1
and this is the smallest sequence numbered packet for which the
receiver makes a mistake.  Let j be the packet that the receiver is
currently expecting to receive.  Again, j<i, j=i or j>i.

j<=i: the receiver has not yet sent ACK i since it is expecting packet
j, and cannot buffer ahead.  Hence, j must still be in the sender's
window, such that any packet > j+w-1 cannot yet appear in the window.
Hence, the receiver should know that i+w+1 > j+w-1 cannot yet have been
sent.

j>i: When the sender sent i, its window could not yet have covered
i+w.  Since i+w had not yet been sent, the receiver could not have
ACK'd i+w without having made a mistake.  In other words, the receiver
cannot receive i+w correctly until the sender has stopped sending i.
This has not yet happened, so the receiver must ignore the packet it
thinks is i+w+1, and send ACK j-1.  This is exactly what it would have
done had it thought the packet was indeed i.