fix: add lifespan context manager to StreamableHTTP mounting examples

The ASGI mounting examples were missing the lifespan context manager
needed to properly initialize the session manager. Without this,
requests fail with "RuntimeError: Task group is not initialized".

This adds the proper lifespan pattern to:
- streamable_http_basic_mounting.py
- streamable_http_host_mounting.py
- streamable_http_multiple_servers.py

Github-Issue:#1484

Author: TheMailmans

examples/snippets/servers/streamable_http_basic_mounting.py

@@ -5,6 +5,8 @@
     uvicorn examples.snippets.servers.streamable_http_basic_mounting:app --reload
 """
 
+import contextlib
+
 from starlette.applications import Starlette
 from starlette.routing import Mount
 
@@ -20,9 +22,17 @@ 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,
 )

examples/snippets/servers/streamable_http_host_mounting.py

@@ -5,6 +5,8 @@
     uvicorn examples.snippets.servers.streamable_http_host_mounting:app --reload
 """
 
+import contextlib
+
 from starlette.applications import Starlette
 from starlette.routing import Host
 
@@ -20,9 +22,17 @@ 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,
 )

examples/snippets/servers/streamable_http_multiple_servers.py

@@ -5,6 +5,8 @@
     uvicorn examples.snippets.servers.streamable_http_multiple_servers:app --reload
 """
 
+import contextlib
+
 from starlette.applications import Starlette
 from starlette.routing import Mount
 
@@ -32,10 +34,21 @@ 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,
 )