Update README

Author: Kludex

README.md

@@ -1394,6 +1394,8 @@ Run from the repository root:
     uvicorn examples.snippets.servers.streamable_http_basic_mounting:app --reload
 """
 
+import contextlib
+
 from starlette.applications import Starlette
 from starlette.routing import Mount
 
@@ -1409,11 +1411,19 @@ def hello() -> str:
     return "Hello from MCP!"
 
 
+# Create a lifespan context manager to run the session manager
+@contextlib.asynccontextmanager
+async def lifespan(app: Starlette):
+    async with mcp.session_manager.run():
+        yield
+
+
 # Mount the StreamableHTTP server to the existing ASGI server
 app = Starlette(
     routes=[
         Mount("/", app=mcp.streamable_http_app()),
-    ]
+    ],
+    lifespan=lifespan,
 )
 ```
 
@@ -1431,6 +1441,8 @@ Run from the repository root:
     uvicorn examples.snippets.servers.streamable_http_host_mounting:app --reload
 """
 
+import contextlib
+
 from starlette.applications import Starlette
 from starlette.routing import Host
 
@@ -1446,11 +1458,19 @@ def domain_info() -> str:
     return "This is served from mcp.acme.corp"
 
 
+# Create a lifespan context manager to run the session manager
+@contextlib.asynccontextmanager
+async def lifespan(app: Starlette):
+    async with mcp.session_manager.run():
+        yield
+
+
 # Mount using Host-based routing
 app = Starlette(
     routes=[
         Host("mcp.acme.corp", app=mcp.streamable_http_app()),
-    ]
+    ],
+    lifespan=lifespan,
 )
 ```
 
@@ -1468,6 +1488,8 @@ Run from the repository root:
     uvicorn examples.snippets.servers.streamable_http_multiple_servers:app --reload
 """
 
+import contextlib
+
 from starlette.applications import Starlette
 from starlette.routing import Mount
 
@@ -1495,12 +1517,23 @@ def send_message(message: str) -> str:
 api_mcp.settings.streamable_http_path = "/"
 chat_mcp.settings.streamable_http_path = "/"
 
+
+# Create a combined lifespan to manage both session managers
+@contextlib.asynccontextmanager
+async def lifespan(app: Starlette):
+    async with contextlib.AsyncExitStack() as stack:
+        await stack.enter_async_context(api_mcp.session_manager.run())
+        await stack.enter_async_context(chat_mcp.session_manager.run())
+        yield
+
+
 # Mount the servers
 app = Starlette(
     routes=[
         Mount("/api", app=api_mcp.streamable_http_app()),
         Mount("/chat", app=chat_mcp.streamable_http_app()),
-    ]
+    ],
+    lifespan=lifespan,
 )
 ```