Update README snippets

Author: LucaButBoring

README.md

@@ -502,21 +502,22 @@ cd to the `examples/snippets/clients` directory and run:
     uv run server async_tool_basic stdio
 """
 
-import asyncio
+import anyio
 
 from mcp.server.fastmcp import Context, FastMCP
+from mcp.server.session import ServerSession
 
 mcp = FastMCP("Async Tool Basic")
 
 
 @mcp.tool(invocation_modes=["async"])
-async def analyze_data(dataset: str, ctx: Context) -> str:  # type: ignore[type-arg]
+async def analyze_data(dataset: str, ctx: Context[ServerSession, None]) -> str:
     """Analyze a dataset asynchronously with progress updates."""
     await ctx.info(f"Starting analysis of {dataset}")
 
     # Simulate analysis with progress updates
     for i in range(5):
-        await asyncio.sleep(0.5)
+        await anyio.sleep(0.5)
         progress = (i + 1) / 5
         await ctx.report_progress(progress, 1.0, f"Processing step {i + 1}/5")
 
@@ -525,21 +526,11 @@ async def analyze_data(dataset: str, ctx: Context) -> str:  # type: ignore[type-
 
 
 @mcp.tool(invocation_modes=["sync", "async"])
-def process_text(text: str, ctx: Context | None = None) -> str:  # type: ignore[type-arg]
+async def process_text(text: str, ctx: Context[ServerSession, None]) -> str:
     """Process text in sync or async mode."""
-    if ctx:
-        # Async mode with context
-        import asyncio
 
-        async def async_processing():
-            await ctx.info(f"Processing text asynchronously: {text[:20]}...")
-            await asyncio.sleep(0.3)
-
-        try:
-            loop = asyncio.get_event_loop()
-            loop.create_task(async_processing())
-        except RuntimeError:
-            pass
+    await ctx.info(f"Processing text asynchronously: {text[:20]}...")
+    await anyio.sleep(0.3)
 
     return f"Processed: {text.upper()}"
 
@@ -562,10 +553,11 @@ cd to the `examples/snippets/clients` directory and run:
     uv run server async_tool_immediate stdio
 """
 
-import asyncio
+import anyio
 
 from mcp import types
 from mcp.server.fastmcp import Context, FastMCP
+from mcp.server.session import ServerSession
 
 mcp = FastMCP("Async Tool Immediate")
 
@@ -576,13 +568,13 @@ async def provide_immediate_feedback(operation: str) -> list[types.ContentBlock]
 
 
 @mcp.tool(invocation_modes=["async"], immediate_result=provide_immediate_feedback)
-async def long_analysis(operation: str, ctx: Context) -> str:  # type: ignore[type-arg]
+async def long_analysis(operation: str, ctx: Context[ServerSession, None]) -> str:
     """Perform long-running analysis with immediate user feedback."""
     await ctx.info(f"Beginning {operation} analysis")
 
     # Simulate long-running work
     for i in range(4):
-        await asyncio.sleep(1)
+        await anyio.sleep(1)
         progress = (i + 1) / 4
         await ctx.report_progress(progress, 1.0, f"Analysis step {i + 1}/4")
 
@@ -607,9 +599,10 @@ cd to the `examples/snippets` directory and run:
     uv run async-tool-client
 """
 
-import asyncio
 import os
 
+import anyio
+
 from mcp import ClientSession, StdioServerParameters, types
 from mcp.client.stdio import stdio_client
 
@@ -646,7 +639,7 @@ async def call_async_tool(session: ClientSession):
                 print(f"Operation failed: {status.error}")
                 break
 
-            await asyncio.sleep(0.5)
+            await anyio.sleep(0.5)
 
 
 async def run():
@@ -658,7 +651,7 @@ async def run():
 
 
 if __name__ == "__main__":
-    asyncio.run(run())
+    anyio.run(run)
 ```
 
 _Full example: [examples/snippets/clients/async_tool_client.py](https://github.com/modelcontextprotocol/python-sdk/blob/main/examples/snippets/clients/async_tool_client.py)_