diff --git a/README.md b/README.md
index e323d08..bd5f2fc 100644
--- a/README.md
+++ b/README.md
@@ -58,7 +58,7 @@ func main() {
 			} else {
 				// This shouldn't happen in this example
 				fmt.Println("WARN: Response was not shared!")
-            }
+			}
 
 			// We discard here, but a more realistic handling might be to stream
 			// the response to a user.
diff --git a/seeker.go b/seeker.go
index 3c164e3..46ccc25 100644
--- a/seeker.go
+++ b/seeker.go
@@ -1,6 +1,7 @@
 package sfstreams
 
 import (
+	"errors"
 	"io"
 	"sync"
 )
@@ -42,28 +43,47 @@ type downstreamSeeker struct {
 	io.ReadSeekCloser
 	parent *parentSeeker
 	pos    int64
+	eof    bool
+	eofPos int64
+	closed bool
 }
 
 func newSyncSeeker(parent *parentSeeker) *downstreamSeeker {
 	return &downstreamSeeker{
 		parent: parent,
 		pos:    0,
+		eof:    false,
+		eofPos: 0,
+		closed: false,
 	}
 }
 
 func (s *downstreamSeeker) Read(b []byte) (int, error) {
+	if s.closed {
+		return 0, io.ErrClosedPipe
+	}
 	s.parent.mutex.Lock()
 	defer s.parent.mutex.Unlock()
+	if s.eof && s.pos == s.eofPos {
+		return 0, io.EOF
+	}
 	offset, err := s.parent.Seek(s.pos, io.SeekStart)
 	if err != nil {
 		return 0, err
 	}
 	i, err := s.parent.Read(b)
 	s.pos = offset + int64(i)
+	if err != nil && errors.Is(err, io.EOF) {
+		s.eof = true
+		s.eofPos = s.pos
+	}
 	return i, err
 }
 
 func (s *downstreamSeeker) Seek(offset int64, whence int) (int64, error) {
+	if s.closed {
+		return 0, io.ErrClosedPipe
+	}
 	s.parent.mutex.Lock()
 	defer s.parent.mutex.Unlock()
 	offset, err := s.parent.Seek(offset, whence)
@@ -76,5 +96,6 @@ func (s *downstreamSeeker) Seek(offset int64, whence int) (int64, error) {
 
 func (s *downstreamSeeker) Close() error {
 	s.parent.closeWg.Done()
+	s.closed = true
 	return nil
 }
diff --git a/seeker_test.go b/seeker_test.go
new file mode 100644
index 0000000..17faa35
--- /dev/null
+++ b/seeker_test.go
@@ -0,0 +1,166 @@
+package sfstreams
+
+import (
+	"bytes"
+	"crypto/rand"
+	"errors"
+	"io"
+	"testing"
+)
+
+type bufCloser struct {
+	io.ReadSeeker
+	io.Closer
+}
+
+func (b *bufCloser) Close() error {
+	return nil // no-op
+}
+
+func createSource(length int64, t *testing.T) (io.ReadSeekCloser, []byte) {
+	buf := make([]byte, length)
+	i, err := rand.Read(buf)
+	if err != nil {
+		panic(err)
+	}
+	if int64(i) != length {
+		t.Fatal("did not read enough random bytes")
+	}
+	return &bufCloser{ReadSeeker: bytes.NewReader(buf)}, buf
+}
+
+func TestDuplicateReads(t *testing.T) {
+	rsc, b := createSource(1024, t)
+	ps := newParentSeeker(rsc, 2)
+	s1 := newSyncSeeker(ps)
+	s2 := newSyncSeeker(ps)
+
+	// Seek to different places
+	_, err := s1.Seek(512, io.SeekStart)
+	if err != nil {
+		t.Fatal(err)
+	}
+	_, err = s2.Seek(128, io.SeekStart)
+	if err != nil {
+		t.Fatal(err)
+	}
+
+	// Read a segment of bytes from each
+	b1 := make([]byte, 128)
+	b2 := make([]byte, 128)
+	_, err = s1.Read(b1)
+	if err != nil {
+		t.Fatal(err)
+	}
+	_, err = s2.Read(b2)
+	if err != nil {
+		t.Fatal(err)
+	}
+
+	// Compare each segment to ensure the correct thing was read
+	for i := 512; i < 640; i++ {
+		if b1[i-512] != b[i] {
+			t.Fatalf("byte %d in segment 1 is incorrect", i)
+		}
+	}
+	for i := 128; i < 256; i++ {
+		if b2[i-128] != b[i] {
+			t.Fatalf("byte %d in segment 2 is incorrect", i)
+		}
+	}
+}
+
+func TestOverRead(t *testing.T) {
+	rsc, _ := createSource(1024, t)
+	ps := newParentSeeker(rsc, 1)
+	s1 := newSyncSeeker(ps)
+
+	// Discard the whole stream
+	_, err := io.Copy(io.Discard, s1)
+	if err != nil {
+		t.Fatal(err)
+	}
+
+	// Read from it again
+	b := make([]byte, 128)
+	i, err := s1.Read(b)
+	if !errors.Is(err, io.EOF) {
+		t.Fatal(err)
+	}
+	if i > 0 {
+		t.Fatalf("expected to read zero bytes, got %d", i)
+	}
+}
+
+type badStream struct {
+	io.ReadSeekCloser
+	pos    int64
+	source *bytes.Reader
+}
+
+func (b *badStream) Read(buf []byte) (int, error) {
+	if b.pos >= int64(b.source.Len()) {
+		return 0, errors.New("the requested range cannot be satisfied")
+	}
+	i, err := b.source.Read(buf)
+	if !errors.Is(err, io.EOF) && (int64(i)+b.pos) >= int64(b.source.Len()) {
+		return i, io.EOF
+	}
+	return i, err
+}
+
+func (b *badStream) Seek(offset int64, whence int) (int64, error) {
+	p, err := b.source.Seek(offset, whence)
+	b.pos = p
+	return p, err
+}
+
+func (b *badStream) Close() error {
+	return nil // no-op
+}
+
+func TestImproperSourceOverRead(t *testing.T) {
+	_, b := createSource(1024, t)
+	bs := &badStream{source: bytes.NewReader(b)}
+	ps := newParentSeeker(bs, 1)
+	s1 := newSyncSeeker(ps)
+
+	// Discard the whole stream
+	_, err := io.Copy(io.Discard, s1)
+	if err != nil {
+		t.Fatal(err)
+	}
+
+	// Read from it again
+	rb := make([]byte, 128)
+	i, err := s1.Read(rb)
+	if !errors.Is(err, io.EOF) {
+		t.Fatal(err)
+	}
+	if i > 0 {
+		t.Fatalf("expected to read zero bytes, got %d", i)
+	}
+}
+
+func TestUseAfterClose(t *testing.T) {
+	rsc, _ := createSource(1024, t)
+	ps := newParentSeeker(rsc, 1)
+	s1 := newSyncSeeker(ps)
+
+	// Close the whole thing
+	err := s1.Close()
+	if err != nil {
+		t.Fatal(err)
+	}
+
+	// Now try to read/seek from it
+	b := make([]byte, 128)
+	_, err = s1.Read(b)
+	if !errors.Is(err, io.ErrClosedPipe) {
+		t.Fatal(err)
+	}
+	_, err = s1.Seek(12, io.SeekStart)
+	if !errors.Is(err, io.ErrClosedPipe) {
+		t.Fatal(err)
+	}
+}