fix: add lifespan context manager to StreamableHTTP mounting examples (#1669)

Co-authored-by: TheMailmans <tyler@example.com>
Co-authored-by: Marcelo Trylesinski <marcelotryle@gmail.com>

examples/snippets/servers/streamable_http_multiple_servers.py

@@ -5,6 +5,8 @@ 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
 
@@ -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,
 )